G := PSU(3,8); involutions := [ [ G!(1, 3)(2, 276)(4, 310)(5, 90)(6, 152)(7, 407)(8, 244)(9, 436)(10, 203)(11, 17)(12, 287)(13, 281)(14, 24)(15, 435)(16, 230)(18, 432)(19, 242)(20, 258)(21, 444)(22, 262)(23, 412)(25, 43)(26, 443)(27, 223)(28, 413)(29, 229)(30, 509)(31, 207)(32, 332)(33, 374)(34, 505)(35, 198)(36, 403)(37, 342)(38, 73)(39, 263)(40, 157)(41, 167)(42, 114)(44, 463)(45, 370)(46, 124)(47, 309)(48, 400)(49, 189)(50, 365)(51, 296)(52, 259)(53, 503)(54, 380)(55, 193)(56, 151)(57, 404)(58, 171)(59, 387)(60, 350)(61, 213)(62, 168)(63, 302)(64, 331)(65, 154)(66, 126)(67, 426)(68, 314)(69, 159)(70, 192)(71, 369)(72, 502)(74, 318)(75, 191)(76, 347)(77, 345)(78, 81)(79, 508)(80, 420)(82, 228)(83, 268)(84, 424)(85, 377)(86, 304)(87, 227)(88, 442)(89, 480)(91, 135)(92, 297)(93, 301)(94, 352)(95, 122)(96, 512)(97, 305)(98, 226)(99, 292)(100, 355)(101, 109)(102, 449)(103, 421)(104, 438)(105, 178)(106, 295)(107, 143)(108, 272)(110, 251)(111, 327)(112, 288)(113, 282)(115, 235)(116, 368)(117, 121)(118, 349)(119, 252)(120, 140)(123, 359)(125, 286)(127, 357)(128, 216)(129, 351)(130, 410)(131, 273)(132, 485)(133, 138)(134, 205)(136, 378)(137, 358)(139, 418)(141, 371)(142, 494)(144, 409)(145, 166)(146, 406)(147, 362)(148, 261)(149, 462)(150, 241)(153, 471)(155, 284)(156, 169)(158, 496)(160, 513)(161, 319)(162, 212)(163, 201)(164, 197)(165, 190)(170, 209)(172, 506)(173, 256)(174, 267)(175, 390)(176, 285)(177, 339)(179, 441)(180, 194)(181, 219)(182, 434)(183, 510)(184, 504)(185, 298)(186, 495)(187, 250)(188, 478)(195, 299)(196, 202)(199, 335)(200, 364)(204, 348)(206, 497)(208, 423)(210, 500)(211, 329)(214, 275)(215, 266)(217, 405)(218, 312)(220, 391)(221, 253)(222, 451)(224, 321)(225, 461)(231, 334)(232, 317)(234, 431)(236, 237)(238, 270)(239, 433)(240, 470)(243, 392)(245, 338)(246, 456)(247, 408)(248, 307)(249, 469)(254, 360)(255, 428)(257, 356)(260, 381)(264, 308)(265, 454)(269, 344)(271, 306)(274, 459)(277, 343)(278, 448)(279, 303)(280, 354)(283, 422)(289, 366)(290, 511)(291, 483)(293, 316)(294, 417)(300, 383)(311, 425)(313, 452)(315, 468)(320, 464)(322, 353)(323, 482)(324, 427)(325, 486)(326, 499)(328, 397)(330, 361)(333, 382)(336, 490)(337, 394)(340, 476)(341, 446)(346, 430)(363, 376)(367, 389)(372, 375)(373, 453)(379, 507)(384, 411)(385, 437)(386, 388)(393, 447)(395, 472)(396, 457)(398, 498)(399, 492)(401, 429)(402, 474)(414, 484)(415, 489)(416, 473)(419, 460)(439, 481)(440, 479)(445, 465)(450, 487)(455, 466)(458, 488)(467, 501)(475, 493)(477, 491), G!(1, 272)(2, 218)(3, 349)(4, 203)(5, 164)(6, 259)(7, 488)(8, 409)(9, 304)(10, 267)(11, 194)(12, 352)(13, 314)(14, 322)(15, 353)(16, 77)(17, 312)(18, 483)(19, 439)(20, 99)(21, 512)(22, 138)(23, 318)(24, 68)(25, 324)(26, 35)(27, 222)(28, 377)(29, 206)(30, 223)(31, 173)(32, 41)(33, 252)(34, 105)(36, 152)(37, 417)(38, 413)(39, 447)(40, 340)(42, 170)(43, 117)(44, 320)(45, 260)(46, 500)(47, 250)(48, 130)(49, 289)(50, 435)(51, 341)(52, 404)(53, 376)(54, 385)(55, 351)(56, 169)(57, 490)(58, 276)(59, 459)(60, 469)(61, 178)(62, 200)(63, 236)(64, 270)(65, 295)(66, 350)(67, 356)(69, 415)(70, 450)(71, 426)(72, 327)(73, 147)(74, 378)(75, 354)(76, 456)(78, 104)(79, 489)(80, 369)(81, 416)(82, 344)(83, 454)(84, 497)(85, 422)(86, 392)(87, 166)(88, 410)(89, 242)(90, 299)(91, 208)(92, 366)(93, 125)(94, 465)(95, 419)(96, 437)(97, 315)(98, 429)(100, 499)(101, 275)(102, 400)(103, 484)(106, 371)(107, 449)(108, 381)(109, 246)(110, 241)(111, 334)(112, 183)(113, 172)(114, 464)(115, 254)(116, 504)(118, 127)(119, 197)(120, 438)(121, 364)(122, 226)(123, 268)(124, 412)(126, 233)(128, 224)(129, 507)(131, 142)(132, 323)(133, 431)(134, 339)(135, 245)(136, 389)(137, 402)(139, 394)(140, 408)(141, 159)(143, 387)(144, 346)(145, 207)(146, 210)(148, 294)(149, 176)(150, 494)(151, 227)(153, 237)(154, 175)(155, 179)(156, 496)(157, 301)(158, 256)(160, 436)(161, 287)(162, 243)(163, 277)(165, 190)(167, 306)(168, 285)(171, 187)(174, 397)(177, 491)(180, 309)(181, 319)(182, 460)(184, 235)(185, 307)(186, 196)(188, 330)(189, 463)(191, 493)(192, 503)(193, 420)(195, 373)(198, 403)(199, 308)(201, 359)(202, 355)(204, 357)(205, 466)(209, 297)(211, 247)(212, 321)(213, 291)(214, 234)(215, 238)(216, 513)(217, 333)(219, 335)(220, 479)(221, 271)(225, 383)(228, 343)(229, 328)(230, 448)(231, 360)(232, 281)(239, 302)(240, 375)(244, 391)(248, 425)(251, 361)(253, 298)(255, 273)(257, 379)(258, 510)(261, 433)(262, 347)(263, 432)(264, 445)(265, 269)(266, 292)(274, 442)(278, 481)(279, 380)(280, 511)(282, 495)(283, 458)(284, 498)(286, 386)(288, 331)(290, 501)(293, 452)(296, 467)(300, 430)(303, 485)(305, 423)(310, 424)(311, 332)(313, 468)(316, 338)(317, 365)(325, 382)(326, 395)(329, 411)(336, 443)(337, 405)(342, 471)(345, 396)(348, 370)(358, 401)(362, 407)(363, 451)(367, 406)(368, 502)(372, 418)(374, 399)(384, 473)(388, 414)(390, 508)(393, 505)(398, 455)(421, 476)(427, 462)(428, 478)(434, 474)(440, 461)(441, 477)(444, 482)(446, 475)(453, 492)(457, 480)(470, 486)(472, 506)(487, 509), G!(1, 334)(2, 459)(3, 343)(4, 431)(5, 128)(6, 446)(7, 342)(8, 143)(9, 75)(10, 187)(11, 181)(12, 95)(13, 130)(14, 356)(15, 475)(16, 117)(17, 402)(18, 63)(19, 245)(20, 326)(21, 401)(22, 415)(23, 335)(24, 212)(25, 65)(26, 186)(27, 165)(28, 425)(29, 133)(30, 286)(31, 490)(32, 220)(33, 421)(34, 333)(35, 240)(36, 38)(37, 359)(39, 482)(40, 348)(41, 484)(42, 498)(43, 398)(44, 138)(45, 308)(46, 76)(47, 52)(48, 423)(49, 465)(50, 255)(51, 174)(53, 273)(54, 505)(55, 504)(56, 120)(57, 439)(58, 231)(59, 232)(60, 382)(61, 414)(62, 360)(64, 291)(66, 288)(67, 99)(68, 141)(69, 144)(70, 77)(71, 247)(72, 438)(73, 338)(74, 301)(78, 394)(79, 127)(80, 151)(81, 219)(82, 441)(83, 384)(84, 424)(85, 473)(86, 311)(87, 157)(88, 430)(89, 256)(90, 364)(91, 443)(92, 403)(93, 451)(94, 176)(96, 511)(97, 129)(98, 500)(100, 432)(101, 449)(102, 494)(103, 447)(104, 287)(105, 279)(106, 270)(107, 506)(108, 321)(109, 304)(110, 485)(111, 222)(112, 349)(113, 412)(114, 213)(115, 266)(116, 358)(118, 192)(119, 197)(121, 416)(122, 300)(123, 211)(124, 407)(125, 477)(126, 471)(131, 379)(132, 463)(134, 344)(135, 314)(136, 355)(137, 202)(139, 458)(140, 467)(142, 395)(145, 499)(146, 354)(147, 225)(148, 184)(149, 169)(150, 404)(152, 336)(153, 248)(154, 167)(155, 429)(156, 374)(158, 436)(159, 397)(160, 161)(162, 285)(163, 428)(164, 298)(166, 272)(168, 332)(170, 274)(171, 276)(172, 392)(173, 305)(175, 265)(177, 196)(178, 385)(179, 237)(180, 488)(182, 323)(183, 442)(185, 218)(188, 350)(190, 378)(191, 399)(193, 376)(194, 417)(195, 238)(198, 457)(199, 492)(200, 411)(201, 366)(203, 226)(204, 497)(205, 383)(206, 470)(207, 491)(208, 275)(209, 277)(210, 503)(214, 254)(215, 426)(216, 469)(217, 302)(221, 513)(223, 405)(224, 512)(227, 396)(228, 372)(229, 476)(230, 262)(233, 386)(234, 420)(235, 479)(236, 434)(239, 453)(241, 318)(242, 283)(243, 362)(244, 315)(246, 489)(249, 460)(250, 507)(251, 389)(252, 306)(253, 351)(257, 433)(258, 290)(259, 474)(260, 380)(261, 322)(263, 375)(264, 408)(267, 462)(268, 419)(269, 502)(271, 406)(278, 289)(280, 312)(281, 369)(282, 496)(284, 418)(292, 435)(293, 510)(294, 444)(295, 330)(296, 393)(297, 455)(299, 303)(307, 345)(309, 337)(310, 464)(313, 390)(316, 454)(317, 353)(319, 480)(320, 501)(324, 347)(325, 370)(327, 481)(328, 368)(329, 427)(331, 448)(339, 409)(340, 367)(341, 509)(346, 452)(352, 373)(357, 371)(361, 437)(363, 387)(365, 483)(377, 456)(381, 487)(388, 445)(391, 400)(410, 486)(413, 472)(422, 466)(440, 493)(450, 478)(461, 468)(495, 508) ] , [ G!(1, 258)(2, 228)(3, 333)(4, 130)(5, 214)(6, 509)(7, 422)(8, 439)(9, 483)(10, 247)(11, 125)(12, 248)(13, 270)(14, 374)(15, 348)(16, 398)(17, 205)(18, 220)(19, 457)(20, 382)(21, 163)(22, 467)(23, 407)(24, 138)(25, 307)(26, 364)(27, 361)(28, 416)(29, 353)(30, 302)(31, 365)(32, 171)(33, 133)(34, 511)(35, 237)(36, 106)(37, 489)(38, 101)(39, 443)(40, 249)(41, 208)(42, 340)(43, 287)(44, 103)(45, 317)(46, 337)(47, 83)(48, 254)(49, 143)(50, 159)(51, 239)(52, 227)(53, 295)(54, 99)(55, 262)(56, 109)(57, 438)(58, 158)(59, 380)(60, 504)(61, 376)(62, 442)(63, 152)(64, 113)(65, 461)(66, 379)(67, 260)(68, 300)(69, 207)(70, 257)(71, 240)(72, 201)(73, 151)(74, 219)(75, 105)(76, 354)(77, 224)(78, 242)(79, 360)(80, 334)(81, 396)(82, 464)(84, 184)(85, 281)(86, 405)(87, 392)(88, 311)(89, 347)(90, 349)(91, 383)(92, 426)(93, 127)(94, 172)(95, 126)(96, 114)(97, 225)(98, 326)(100, 418)(102, 218)(104, 123)(107, 327)(108, 192)(110, 420)(111, 189)(112, 322)(115, 117)(116, 484)(118, 275)(119, 429)(120, 312)(121, 253)(122, 507)(124, 454)(128, 129)(131, 357)(132, 482)(134, 286)(135, 314)(136, 150)(137, 236)(139, 500)(140, 449)(141, 296)(142, 250)(144, 266)(145, 160)(146, 195)(147, 408)(148, 256)(149, 166)(153, 277)(154, 305)(155, 303)(156, 335)(157, 417)(161, 318)(162, 352)(164, 428)(165, 268)(167, 271)(168, 425)(169, 316)(170, 174)(173, 367)(175, 391)(176, 269)(177, 241)(178, 488)(179, 226)(180, 351)(181, 319)(182, 487)(183, 299)(185, 463)(186, 436)(187, 474)(188, 369)(190, 309)(191, 458)(193, 501)(194, 216)(196, 445)(197, 451)(198, 358)(199, 293)(200, 263)(202, 395)(203, 362)(204, 493)(206, 481)(209, 452)(210, 355)(211, 491)(212, 506)(213, 453)(215, 246)(217, 448)(221, 235)(222, 255)(223, 411)(229, 288)(230, 399)(231, 251)(232, 440)(233, 328)(234, 342)(238, 377)(243, 259)(244, 497)(245, 321)(252, 264)(261, 389)(265, 394)(267, 313)(272, 356)(273, 301)(274, 343)(276, 320)(278, 304)(279, 486)(280, 480)(282, 385)(283, 412)(284, 325)(285, 460)(289, 477)(290, 468)(291, 495)(292, 387)(294, 469)(297, 381)(298, 421)(306, 423)(308, 401)(310, 388)(315, 505)(323, 466)(324, 473)(329, 366)(330, 384)(331, 437)(332, 496)(336, 414)(338, 345)(339, 378)(341, 375)(344, 419)(346, 372)(350, 424)(359, 404)(363, 373)(368, 490)(370, 479)(371, 433)(386, 410)(390, 432)(393, 434)(400, 508)(402, 494)(403, 503)(406, 510)(409, 456)(413, 427)(415, 431)(430, 446)(435, 475)(441, 499)(444, 502)(447, 450)(455, 485)(459, 471)(462, 513)(465, 472)(470, 478)(476, 512)(492, 498), G!(1, 443)(2, 103)(3, 138)(4, 348)(5, 209)(6, 229)(7, 50)(8, 147)(9, 207)(10, 391)(11, 47)(12, 412)(13, 251)(14, 389)(15, 54)(16, 201)(17, 303)(18, 142)(19, 506)(20, 125)(21, 45)(22, 75)(23, 255)(24, 322)(25, 306)(26, 492)(27, 60)(28, 46)(29, 470)(30, 171)(31, 317)(32, 33)(34, 384)(35, 48)(36, 118)(37, 345)(38, 98)(39, 297)(40, 467)(41, 158)(42, 113)(43, 500)(44, 329)(49, 183)(51, 279)(52, 274)(53, 218)(55, 333)(56, 124)(57, 212)(58, 410)(59, 102)(61, 104)(62, 92)(63, 390)(64, 351)(65, 127)(66, 461)(67, 474)(68, 136)(69, 239)(70, 267)(71, 246)(72, 493)(73, 503)(74, 248)(76, 174)(77, 311)(78, 110)(79, 88)(80, 132)(81, 399)(82, 456)(83, 422)(84, 438)(85, 175)(86, 227)(87, 249)(89, 216)(90, 346)(91, 341)(93, 204)(94, 260)(95, 194)(96, 488)(97, 126)(99, 357)(100, 215)(101, 240)(105, 159)(106, 394)(107, 486)(108, 238)(109, 476)(111, 316)(112, 457)(114, 402)(115, 182)(116, 407)(117, 377)(119, 200)(120, 330)(121, 198)(122, 448)(123, 177)(128, 195)(129, 353)(130, 453)(131, 466)(133, 281)(134, 173)(135, 375)(137, 143)(139, 155)(140, 417)(141, 473)(144, 153)(145, 261)(146, 452)(148, 298)(149, 220)(150, 469)(151, 325)(152, 295)(154, 454)(156, 440)(157, 434)(160, 471)(161, 191)(162, 224)(163, 436)(164, 447)(165, 323)(167, 262)(168, 210)(169, 352)(170, 199)(172, 308)(176, 231)(178, 241)(179, 371)(180, 336)(181, 187)(184, 271)(185, 233)(186, 400)(188, 497)(189, 367)(190, 369)(192, 409)(193, 509)(196, 459)(197, 366)(202, 221)(203, 386)(205, 483)(206, 449)(208, 380)(211, 406)(213, 259)(214, 320)(217, 270)(219, 490)(222, 494)(223, 319)(225, 343)(226, 372)(228, 455)(230, 418)(232, 284)(234, 411)(235, 355)(236, 393)(237, 299)(242, 462)(243, 373)(244, 450)(245, 275)(247, 463)(250, 419)(252, 256)(253, 424)(254, 481)(257, 361)(258, 468)(263, 504)(264, 388)(265, 403)(266, 401)(268, 363)(269, 364)(272, 309)(273, 482)(276, 489)(277, 342)(278, 460)(280, 292)(282, 415)(283, 445)(285, 293)(286, 370)(287, 327)(288, 313)(289, 480)(290, 413)(291, 328)(294, 310)(296, 405)(300, 442)(301, 356)(302, 495)(304, 433)(305, 425)(307, 315)(312, 446)(314, 379)(318, 444)(321, 335)(324, 347)(326, 439)(331, 510)(332, 359)(334, 428)(337, 340)(338, 478)(339, 365)(344, 397)(349, 354)(350, 496)(358, 511)(360, 479)(362, 432)(368, 374)(376, 484)(378, 423)(381, 499)(382, 392)(383, 398)(385, 507)(387, 464)(395, 472)(396, 408)(404, 430)(414, 458)(416, 502)(420, 441)(421, 426)(427, 513)(429, 501)(431, 498)(435, 437)(451, 491)(465, 512)(475, 485)(477, 487)(505, 508), G!(1, 401)(2, 112)(3, 377)(4, 71)(5, 347)(6, 236)(7, 274)(8, 23)(9, 69)(10, 504)(11, 245)(12, 213)(13, 459)(14, 54)(15, 355)(16, 333)(17, 125)(18, 299)(19, 139)(20, 67)(21, 423)(22, 419)(24, 414)(25, 86)(26, 373)(27, 247)(28, 468)(29, 369)(30, 387)(31, 95)(32, 463)(33, 326)(34, 41)(35, 466)(36, 185)(37, 155)(38, 444)(39, 70)(40, 409)(42, 457)(43, 371)(44, 456)(45, 310)(46, 474)(47, 168)(48, 334)(49, 153)(50, 348)(51, 56)(52, 439)(53, 388)(55, 325)(57, 332)(58, 128)(59, 248)(60, 188)(61, 462)(62, 427)(63, 234)(64, 293)(65, 290)(66, 467)(68, 345)(72, 147)(73, 116)(74, 244)(75, 426)(76, 471)(77, 190)(78, 379)(79, 323)(80, 199)(81, 179)(82, 228)(83, 261)(84, 170)(85, 350)(87, 302)(88, 359)(89, 109)(90, 381)(91, 446)(92, 422)(93, 131)(94, 406)(96, 191)(97, 169)(98, 123)(99, 421)(100, 287)(101, 300)(102, 412)(103, 142)(104, 416)(105, 145)(106, 113)(107, 385)(108, 178)(110, 503)(111, 469)(114, 490)(115, 395)(117, 484)(118, 331)(119, 273)(120, 317)(121, 432)(122, 306)(124, 498)(126, 259)(127, 314)(129, 223)(130, 349)(132, 164)(133, 156)(134, 319)(135, 182)(136, 221)(137, 362)(138, 343)(140, 361)(141, 262)(143, 413)(144, 215)(146, 189)(148, 154)(149, 405)(150, 276)(151, 454)(152, 183)(157, 162)(158, 201)(159, 477)(160, 512)(161, 376)(163, 229)(165, 431)(166, 480)(167, 408)(171, 177)(172, 312)(173, 358)(174, 353)(175, 448)(176, 435)(180, 464)(181, 458)(184, 264)(186, 380)(187, 370)(192, 265)(193, 224)(194, 268)(195, 495)(196, 404)(197, 445)(198, 407)(200, 280)(202, 375)(203, 249)(204, 410)(205, 284)(206, 372)(207, 238)(208, 226)(209, 363)(210, 346)(211, 450)(212, 267)(214, 356)(216, 251)(217, 285)(218, 294)(219, 327)(220, 298)(222, 389)(225, 341)(227, 357)(230, 307)(231, 324)(232, 465)(233, 476)(235, 283)(237, 481)(239, 242)(240, 403)(241, 315)(243, 263)(246, 497)(250, 496)(252, 449)(253, 436)(254, 505)(255, 339)(256, 433)(257, 304)(258, 311)(260, 352)(266, 447)(269, 383)(270, 494)(271, 292)(272, 344)(275, 460)(277, 286)(278, 329)(279, 297)(281, 374)(282, 400)(288, 437)(289, 360)(291, 320)(295, 309)(296, 384)(301, 318)(303, 420)(305, 338)(308, 488)(313, 330)(316, 492)(321, 506)(322, 393)(328, 336)(335, 472)(337, 390)(340, 364)(342, 424)(351, 366)(365, 399)(367, 487)(368, 442)(378, 482)(382, 473)(386, 428)(391, 507)(392, 411)(394, 451)(396, 398)(397, 402)(415, 417)(418, 502)(425, 493)(429, 430)(434, 443)(438, 509)(440, 461)(441, 470)(452, 485)(453, 479)(455, 499)(475, 513)(478, 500)(483, 511)(486, 489)(491, 508)(501, 510) ] ]; geometries := [* *]; for i in [1..#involutions] do G1 := sub; G2 := sub; G3 := sub; Append(~geometries,CosetGeometry(G,{G1,G2,G3})); end for;