G := sub<Sym(256)|(2, 4)(3, 6)(5, 10)(7, 14)(8, 16)(9, 18)(11, 22)(12, 24)(13, 26)(15, 30)(17, 34)(19, 38)(20, 40)(21, 42)(23, 46)(25, 50)(27, 54)(28, 56)(29, 
      58)(31, 62)(32, 64)(33, 66)(35, 70)(36, 72)(37, 74)(39, 78)(41, 82)(43, 86)(44, 88)(45, 90)(47, 94)(48, 96)(49, 98)(51, 102)(52, 104)(53, 
      106)(55, 110)(57, 114)(59, 118)(60, 120)(61, 122)(63, 126)(65, 130)(67, 107)(68, 132)(69, 108)(71, 124)(73, 139)(75, 84)(76, 143)(77, 
      144)(79, 92)(80, 85)(81, 149)(83, 153)(87, 158)(89, 162)(91, 166)(93, 111)(95, 101)(97, 113)(99, 173)(100, 175)(103, 125)(105, 181)(109, 
      127)(112, 167)(115, 187)(116, 189)(117, 190)(119, 160)(121, 195)(123, 196)(129, 193)(131, 204)(133, 164)(134, 206)(135, 208)(136, 151)(137, 
      210)(138, 212)(140, 191)(141, 214)(142, 215)(145, 178)(146, 152)(147, 218)(148, 217)(150, 220)(154, 168)(155, 170)(156, 203)(157, 219)(159, 
      198)(161, 211)(163, 226)(165, 228)(169, 188)(171, 232)(172, 224)(174, 176)(177, 235)(179, 236)(180, 238)(183, 184)(185, 240)(186, 213)(192, 
      194)(197, 244)(199, 234)(200, 239)(201, 229)(202, 216)(205, 233)(207, 247)(209, 242)(221, 223)(222, 249)(225, 241)(227, 246)(230, 245)(237, 
      243)(251, 255)(252, 254),
  (1, 2)(3, 7)(4, 8)(5, 11)(6, 12)(9, 19)(10, 20)(13, 27)(14, 28)(15, 31)(16, 32)(17, 35)(18, 36)(21, 43)(22, 44)(23, 47)(24, 48)(25, 51)(26, 
      52)(29, 59)(30, 60)(33, 67)(34, 68)(37, 75)(38, 76)(39, 79)(40, 80)(41, 83)(42, 84)(45, 91)(46, 92)(49, 99)(50, 100)(53, 107)(54, 108)(55, 
      111)(56, 112)(57, 115)(58, 116)(61, 123)(62, 124)(63, 127)(64, 128)(65, 121)(66, 131)(69, 134)(70, 135)(71, 78)(72, 137)(73, 140)(74, 
      141)(77, 145)(81, 150)(82, 151)(85, 103)(86, 156)(87, 159)(88, 160)(89, 163)(90, 164)(93, 169)(94, 120)(95, 170)(96, 154)(97, 161)(98, 
      171)(101, 177)(102, 178)(104, 114)(105, 165)(106, 132)(109, 183)(110, 184)(113, 148)(117, 155)(118, 191)(119, 193)(122, 142)(125, 129)(126, 
      199)(130, 202)(133, 175)(136, 189)(138, 213)(139, 153)(143, 216)(144, 166)(146, 181)(147, 219)(149, 186)(152, 180)(157, 224)(158, 190)(162, 
      214)(167, 229)(168, 201)(172, 215)(173, 233)(174, 234)(176, 188)(179, 237)(182, 217)(185, 241)(187, 242)(192, 207)(194, 243)(195, 210)(196, 
      222)(197, 245)(198, 246)(200, 231)(203, 226)(204, 208)(205, 212)(206, 209)(211, 248)(218, 251)(220, 232)(221, 235)(223, 227)(225, 252)(228, 
      247)(230, 253)(236, 238)(239, 244)(240, 250)(249, 255)(254, 256),
  (1, 3)(2, 5)(4, 9)(6, 13)(7, 15)(8, 17)(10, 21)(11, 23)(12, 25)(14, 29)(16, 33)(18, 37)(19, 39)(20, 41)(22, 45)(24, 49)(26, 53)(27, 55)(28, 
      57)(30, 61)(31, 63)(32, 65)(34, 69)(35, 71)(36, 73)(38, 77)(40, 81)(42, 85)(43, 87)(44, 89)(46, 93)(47, 95)(48, 97)(50, 101)(51, 103)(52, 
      105)(54, 109)(56, 113)(58, 117)(59, 119)(60, 121)(62, 125)(64, 129)(66, 114)(67, 94)(68, 133)(70, 136)(72, 138)(74, 76)(75, 142)(78, 
      146)(79, 147)(80, 148)(82, 152)(83, 154)(84, 155)(86, 157)(88, 161)(90, 165)(91, 167)(92, 168)(96, 108)(98, 172)(99, 174)(100, 176)(102, 
      179)(104, 180)(106, 182)(107, 140)(110, 185)(111, 116)(112, 186)(115, 188)(118, 192)(120, 194)(122, 164)(123, 197)(124, 198)(126, 139)(127, 
      200)(128, 201)(130, 203)(131, 145)(132, 205)(134, 207)(135, 209)(137, 211)(141, 196)(143, 217)(144, 183)(149, 184)(150, 221)(151, 222)(153, 
      223)(156, 210)(158, 225)(159, 166)(160, 173)(162, 193)(163, 227)(169, 230)(170, 231)(171, 216)(175, 195)(177, 214)(178, 218)(181, 239)(187, 
      199)(189, 202)(190, 233)(191, 224)(204, 220)(206, 237)(208, 248)(212, 236)(213, 249)(215, 250)(219, 244)(226, 251)(228, 232)(229, 242)(234, 
      252)(235, 241)(238, 240)(243, 245)(246, 253)(247, 254)(255, 256)>;

involutions :=
[
[
  G!(2, 4)(3, 6)(5, 10)(7, 14)(8, 16)(9, 18)(11, 22)(12, 24)(13, 26)(15, 30)(17, 34)(19, 38)(20, 40)(21, 42)(23, 46)(25, 50)(27, 54)(28, 56)(29, 
      58)(31, 62)(32, 64)(33, 66)(35, 70)(36, 72)(37, 74)(39, 78)(41, 82)(43, 86)(44, 88)(45, 90)(47, 94)(48, 96)(49, 98)(51, 102)(52, 104)(53, 
      106)(55, 110)(57, 114)(59, 118)(60, 120)(61, 122)(63, 126)(65, 130)(67, 107)(68, 132)(69, 108)(71, 124)(73, 139)(75, 84)(76, 143)(77, 
      144)(79, 92)(80, 85)(81, 149)(83, 153)(87, 158)(89, 162)(91, 166)(93, 111)(95, 101)(97, 113)(99, 173)(100, 175)(103, 125)(105, 181)(109, 
      127)(112, 167)(115, 187)(116, 189)(117, 190)(119, 160)(121, 195)(123, 196)(129, 193)(131, 204)(133, 164)(134, 206)(135, 208)(136, 151)(137, 
      210)(138, 212)(140, 191)(141, 214)(142, 215)(145, 178)(146, 152)(147, 218)(148, 217)(150, 220)(154, 168)(155, 170)(156, 203)(157, 219)(159, 
      198)(161, 211)(163, 226)(165, 228)(169, 188)(171, 232)(172, 224)(174, 176)(177, 235)(179, 236)(180, 238)(183, 184)(185, 240)(186, 213)(192, 
      194)(197, 244)(199, 234)(200, 239)(201, 229)(202, 216)(205, 233)(207, 247)(209, 242)(221, 223)(222, 249)(225, 241)(227, 246)(230, 245)(237, 
      243)(251, 255)(252, 254),
  G!(1, 2)(3, 7)(4, 8)(5, 11)(6, 12)(9, 19)(10, 20)(13, 27)(14, 28)(15, 31)(16, 32)(17, 35)(18, 36)(21, 43)(22, 44)(23, 47)(24, 48)(25, 51)(26, 
      52)(29, 59)(30, 60)(33, 67)(34, 68)(37, 75)(38, 76)(39, 79)(40, 80)(41, 83)(42, 84)(45, 91)(46, 92)(49, 99)(50, 100)(53, 107)(54, 108)(55, 
      111)(56, 112)(57, 115)(58, 116)(61, 123)(62, 124)(63, 127)(64, 128)(65, 121)(66, 131)(69, 134)(70, 135)(71, 78)(72, 137)(73, 140)(74, 
      141)(77, 145)(81, 150)(82, 151)(85, 103)(86, 156)(87, 159)(88, 160)(89, 163)(90, 164)(93, 169)(94, 120)(95, 170)(96, 154)(97, 161)(98, 
      171)(101, 177)(102, 178)(104, 114)(105, 165)(106, 132)(109, 183)(110, 184)(113, 148)(117, 155)(118, 191)(119, 193)(122, 142)(125, 129)(126, 
      199)(130, 202)(133, 175)(136, 189)(138, 213)(139, 153)(143, 216)(144, 166)(146, 181)(147, 219)(149, 186)(152, 180)(157, 224)(158, 190)(162, 
      214)(167, 229)(168, 201)(172, 215)(173, 233)(174, 234)(176, 188)(179, 237)(182, 217)(185, 241)(187, 242)(192, 207)(194, 243)(195, 210)(196, 
      222)(197, 245)(198, 246)(200, 231)(203, 226)(204, 208)(205, 212)(206, 209)(211, 248)(218, 251)(220, 232)(221, 235)(223, 227)(225, 252)(228, 
      247)(230, 253)(236, 238)(239, 244)(240, 250)(249, 255)(254, 256),
  G!(1, 3)(2, 5)(4, 9)(6, 13)(7, 15)(8, 17)(10, 21)(11, 23)(12, 25)(14, 29)(16, 33)(18, 37)(19, 39)(20, 41)(22, 45)(24, 49)(26, 53)(27, 55)(28, 
      57)(30, 61)(31, 63)(32, 65)(34, 69)(35, 71)(36, 73)(38, 77)(40, 81)(42, 85)(43, 87)(44, 89)(46, 93)(47, 95)(48, 97)(50, 101)(51, 103)(52, 
      105)(54, 109)(56, 113)(58, 117)(59, 119)(60, 121)(62, 125)(64, 129)(66, 114)(67, 94)(68, 133)(70, 136)(72, 138)(74, 76)(75, 142)(78, 
      146)(79, 147)(80, 148)(82, 152)(83, 154)(84, 155)(86, 157)(88, 161)(90, 165)(91, 167)(92, 168)(96, 108)(98, 172)(99, 174)(100, 176)(102, 
      179)(104, 180)(106, 182)(107, 140)(110, 185)(111, 116)(112, 186)(115, 188)(118, 192)(120, 194)(122, 164)(123, 197)(124, 198)(126, 139)(127, 
      200)(128, 201)(130, 203)(131, 145)(132, 205)(134, 207)(135, 209)(137, 211)(141, 196)(143, 217)(144, 183)(149, 184)(150, 221)(151, 222)(153, 
      223)(156, 210)(158, 225)(159, 166)(160, 173)(162, 193)(163, 227)(169, 230)(170, 231)(171, 216)(175, 195)(177, 214)(178, 218)(181, 239)(187, 
      199)(189, 202)(190, 233)(191, 224)(204, 220)(206, 237)(208, 248)(212, 236)(213, 249)(215, 250)(219, 244)(226, 251)(228, 232)(229, 242)(234, 
      252)(235, 241)(238, 240)(243, 245)(246, 253)(247, 254)(255, 256)
]

];

geometries := [* *];


for i in [1..#involutions] do
	G1 := sub<G|involutions[i][2],involutions[i][3]>;
	G2 := sub<G|involutions[i][1],involutions[i][3]>;
	G3 := sub<G|involutions[i][1],involutions[i][2]>;
	Append(~geometries,CosetGeometry(G,{G1,G2,G3}));
end for;