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Polytope of Type {8,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,8}*512l
if this polytope has a name.
Group : SmallGroup(512,58342)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 32, 128, 32
Order of s0s1s2 : 8
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,4}*256a, {4,8}*256c
   4-fold quotients : {8,4}*128a, {4,4}*128, {8,4}*128b
   8-fold quotients : {8,4}*64a, {8,4}*64b, {4,4}*64
   16-fold quotients : {4,4}*32, {8,2}*32
   32-fold quotients : {2,4}*16, {4,2}*16
   64-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,129)(  2,130)(  3,132)(  4,131)(  5,133)(  6,134)(  7,136)(  8,135)
(  9,137)( 10,138)( 11,140)( 12,139)( 13,141)( 14,142)( 15,144)( 16,143)
( 17,152)( 18,151)( 19,149)( 20,150)( 21,147)( 22,148)( 23,146)( 24,145)
( 25,160)( 26,159)( 27,157)( 28,158)( 29,155)( 30,156)( 31,154)( 32,153)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,173)( 38,174)( 39,176)( 40,175)
( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)( 48,167)
( 49,192)( 50,191)( 51,189)( 52,190)( 53,187)( 54,188)( 55,186)( 56,185)
( 57,184)( 58,183)( 59,181)( 60,182)( 61,179)( 62,180)( 63,178)( 64,177)
( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)( 72,199)
( 73,201)( 74,202)( 75,204)( 76,203)( 77,205)( 78,206)( 79,208)( 80,207)
( 81,216)( 82,215)( 83,213)( 84,214)( 85,211)( 86,212)( 87,210)( 88,209)
( 89,224)( 90,223)( 91,221)( 92,222)( 93,219)( 94,220)( 95,218)( 96,217)
( 97,233)( 98,234)( 99,236)(100,235)(101,237)(102,238)(103,240)(104,239)
(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)(112,231)
(113,256)(114,255)(115,253)(116,254)(117,251)(118,252)(119,250)(120,249)
(121,248)(122,247)(123,245)(124,246)(125,243)(126,244)(127,242)(128,241)
(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)(264,391)
(265,393)(266,394)(267,396)(268,395)(269,397)(270,398)(271,400)(272,399)
(273,408)(274,407)(275,405)(276,406)(277,403)(278,404)(279,402)(280,401)
(281,416)(282,415)(283,413)(284,414)(285,411)(286,412)(287,410)(288,409)
(289,425)(290,426)(291,428)(292,427)(293,429)(294,430)(295,432)(296,431)
(297,417)(298,418)(299,420)(300,419)(301,421)(302,422)(303,424)(304,423)
(305,448)(306,447)(307,445)(308,446)(309,443)(310,444)(311,442)(312,441)
(313,440)(314,439)(315,437)(316,438)(317,435)(318,436)(319,434)(320,433)
(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)(328,455)
(329,457)(330,458)(331,460)(332,459)(333,461)(334,462)(335,464)(336,463)
(337,472)(338,471)(339,469)(340,470)(341,467)(342,468)(343,466)(344,465)
(345,480)(346,479)(347,477)(348,478)(349,475)(350,476)(351,474)(352,473)
(353,489)(354,490)(355,492)(356,491)(357,493)(358,494)(359,496)(360,495)
(361,481)(362,482)(363,484)(364,483)(365,485)(366,486)(367,488)(368,487)
(369,512)(370,511)(371,509)(372,510)(373,507)(374,508)(375,506)(376,505)
(377,504)(378,503)(379,501)(380,502)(381,499)(382,500)(383,498)(384,497);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,263)(  6,264)(  7,261)(  8,262)
(  9,265)( 10,266)( 11,267)( 12,268)( 13,271)( 14,272)( 15,269)( 16,270)
( 17,274)( 18,273)( 19,276)( 20,275)( 21,280)( 22,279)( 23,278)( 24,277)
( 25,282)( 26,281)( 27,284)( 28,283)( 29,288)( 30,287)( 31,286)( 32,285)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,303)( 38,304)( 39,301)( 40,302)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,295)( 46,296)( 47,293)( 48,294)
( 49,314)( 50,313)( 51,316)( 52,315)( 53,320)( 54,319)( 55,318)( 56,317)
( 57,306)( 58,305)( 59,308)( 60,307)( 61,312)( 62,311)( 63,310)( 64,309)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,343)( 70,344)( 71,341)( 72,342)
( 73,345)( 74,346)( 75,347)( 76,348)( 77,351)( 78,352)( 79,349)( 80,350)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,327)( 86,328)( 87,325)( 88,326)
( 89,329)( 90,330)( 91,331)( 92,332)( 93,335)( 94,336)( 95,333)( 96,334)
( 97,380)( 98,379)( 99,378)(100,377)(101,382)(102,381)(103,384)(104,383)
(105,372)(106,371)(107,370)(108,369)(109,374)(110,373)(111,376)(112,375)
(113,364)(114,363)(115,362)(116,361)(117,366)(118,365)(119,368)(120,367)
(121,356)(122,355)(123,354)(124,353)(125,358)(126,357)(127,360)(128,359)
(129,417)(130,418)(131,419)(132,420)(133,423)(134,424)(135,421)(136,422)
(137,425)(138,426)(139,427)(140,428)(141,431)(142,432)(143,429)(144,430)
(145,434)(146,433)(147,436)(148,435)(149,440)(150,439)(151,438)(152,437)
(153,442)(154,441)(155,444)(156,443)(157,448)(158,447)(159,446)(160,445)
(161,385)(162,386)(163,387)(164,388)(165,391)(166,392)(167,389)(168,390)
(169,393)(170,394)(171,395)(172,396)(173,399)(174,400)(175,397)(176,398)
(177,402)(178,401)(179,404)(180,403)(181,408)(182,407)(183,406)(184,405)
(185,410)(186,409)(187,412)(188,411)(189,416)(190,415)(191,414)(192,413)
(193,502)(194,501)(195,504)(196,503)(197,499)(198,500)(199,497)(200,498)
(201,510)(202,509)(203,512)(204,511)(205,507)(206,508)(207,505)(208,506)
(209,485)(210,486)(211,487)(212,488)(213,484)(214,483)(215,482)(216,481)
(217,493)(218,494)(219,495)(220,496)(221,492)(222,491)(223,490)(224,489)
(225,472)(226,471)(227,470)(228,469)(229,465)(230,466)(231,467)(232,468)
(233,480)(234,479)(235,478)(236,477)(237,473)(238,474)(239,475)(240,476)
(241,455)(242,456)(243,453)(244,454)(245,450)(246,449)(247,452)(248,451)
(249,463)(250,464)(251,461)(252,462)(253,458)(254,457)(255,460)(256,459);;
s2 := (  1, 65)(  2, 66)(  3, 68)(  4, 67)(  5, 70)(  6, 69)(  7, 71)(  8, 72)
(  9, 73)( 10, 74)( 11, 76)( 12, 75)( 13, 78)( 14, 77)( 15, 79)( 16, 80)
( 17, 82)( 18, 81)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 88)( 24, 87)
( 25, 90)( 26, 89)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 96)( 32, 95)
( 33,101)( 34,102)( 35,104)( 36,103)( 37, 97)( 38, 98)( 39,100)( 40, 99)
( 41,109)( 42,110)( 43,112)( 44,111)( 45,105)( 46,106)( 47,108)( 48,107)
( 49,117)( 50,118)( 51,120)( 52,119)( 53,113)( 54,114)( 55,116)( 56,115)
( 57,125)( 58,126)( 59,128)( 60,127)( 61,121)( 62,122)( 63,124)( 64,123)
(129,193)(130,194)(131,196)(132,195)(133,198)(134,197)(135,199)(136,200)
(137,201)(138,202)(139,204)(140,203)(141,206)(142,205)(143,207)(144,208)
(145,210)(146,209)(147,211)(148,212)(149,213)(150,214)(151,216)(152,215)
(153,218)(154,217)(155,219)(156,220)(157,221)(158,222)(159,224)(160,223)
(161,229)(162,230)(163,232)(164,231)(165,225)(166,226)(167,228)(168,227)
(169,237)(170,238)(171,240)(172,239)(173,233)(174,234)(175,236)(176,235)
(177,245)(178,246)(179,248)(180,247)(181,241)(182,242)(183,244)(184,243)
(185,253)(186,254)(187,256)(188,255)(189,249)(190,250)(191,252)(192,251)
(257,321)(258,322)(259,324)(260,323)(261,326)(262,325)(263,327)(264,328)
(265,329)(266,330)(267,332)(268,331)(269,334)(270,333)(271,335)(272,336)
(273,338)(274,337)(275,339)(276,340)(277,341)(278,342)(279,344)(280,343)
(281,346)(282,345)(283,347)(284,348)(285,349)(286,350)(287,352)(288,351)
(289,357)(290,358)(291,360)(292,359)(293,353)(294,354)(295,356)(296,355)
(297,365)(298,366)(299,368)(300,367)(301,361)(302,362)(303,364)(304,363)
(305,373)(306,374)(307,376)(308,375)(309,369)(310,370)(311,372)(312,371)
(313,381)(314,382)(315,384)(316,383)(317,377)(318,378)(319,380)(320,379)
(385,449)(386,450)(387,452)(388,451)(389,454)(390,453)(391,455)(392,456)
(393,457)(394,458)(395,460)(396,459)(397,462)(398,461)(399,463)(400,464)
(401,466)(402,465)(403,467)(404,468)(405,469)(406,470)(407,472)(408,471)
(409,474)(410,473)(411,475)(412,476)(413,477)(414,478)(415,480)(416,479)
(417,485)(418,486)(419,488)(420,487)(421,481)(422,482)(423,484)(424,483)
(425,493)(426,494)(427,496)(428,495)(429,489)(430,490)(431,492)(432,491)
(433,501)(434,502)(435,504)(436,503)(437,497)(438,498)(439,500)(440,499)
(441,509)(442,510)(443,512)(444,511)(445,505)(446,506)(447,508)(448,507);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(512)!(  1,129)(  2,130)(  3,132)(  4,131)(  5,133)(  6,134)(  7,136)
(  8,135)(  9,137)( 10,138)( 11,140)( 12,139)( 13,141)( 14,142)( 15,144)
( 16,143)( 17,152)( 18,151)( 19,149)( 20,150)( 21,147)( 22,148)( 23,146)
( 24,145)( 25,160)( 26,159)( 27,157)( 28,158)( 29,155)( 30,156)( 31,154)
( 32,153)( 33,169)( 34,170)( 35,172)( 36,171)( 37,173)( 38,174)( 39,176)
( 40,175)( 41,161)( 42,162)( 43,164)( 44,163)( 45,165)( 46,166)( 47,168)
( 48,167)( 49,192)( 50,191)( 51,189)( 52,190)( 53,187)( 54,188)( 55,186)
( 56,185)( 57,184)( 58,183)( 59,181)( 60,182)( 61,179)( 62,180)( 63,178)
( 64,177)( 65,193)( 66,194)( 67,196)( 68,195)( 69,197)( 70,198)( 71,200)
( 72,199)( 73,201)( 74,202)( 75,204)( 76,203)( 77,205)( 78,206)( 79,208)
( 80,207)( 81,216)( 82,215)( 83,213)( 84,214)( 85,211)( 86,212)( 87,210)
( 88,209)( 89,224)( 90,223)( 91,221)( 92,222)( 93,219)( 94,220)( 95,218)
( 96,217)( 97,233)( 98,234)( 99,236)(100,235)(101,237)(102,238)(103,240)
(104,239)(105,225)(106,226)(107,228)(108,227)(109,229)(110,230)(111,232)
(112,231)(113,256)(114,255)(115,253)(116,254)(117,251)(118,252)(119,250)
(120,249)(121,248)(122,247)(123,245)(124,246)(125,243)(126,244)(127,242)
(128,241)(257,385)(258,386)(259,388)(260,387)(261,389)(262,390)(263,392)
(264,391)(265,393)(266,394)(267,396)(268,395)(269,397)(270,398)(271,400)
(272,399)(273,408)(274,407)(275,405)(276,406)(277,403)(278,404)(279,402)
(280,401)(281,416)(282,415)(283,413)(284,414)(285,411)(286,412)(287,410)
(288,409)(289,425)(290,426)(291,428)(292,427)(293,429)(294,430)(295,432)
(296,431)(297,417)(298,418)(299,420)(300,419)(301,421)(302,422)(303,424)
(304,423)(305,448)(306,447)(307,445)(308,446)(309,443)(310,444)(311,442)
(312,441)(313,440)(314,439)(315,437)(316,438)(317,435)(318,436)(319,434)
(320,433)(321,449)(322,450)(323,452)(324,451)(325,453)(326,454)(327,456)
(328,455)(329,457)(330,458)(331,460)(332,459)(333,461)(334,462)(335,464)
(336,463)(337,472)(338,471)(339,469)(340,470)(341,467)(342,468)(343,466)
(344,465)(345,480)(346,479)(347,477)(348,478)(349,475)(350,476)(351,474)
(352,473)(353,489)(354,490)(355,492)(356,491)(357,493)(358,494)(359,496)
(360,495)(361,481)(362,482)(363,484)(364,483)(365,485)(366,486)(367,488)
(368,487)(369,512)(370,511)(371,509)(372,510)(373,507)(374,508)(375,506)
(376,505)(377,504)(378,503)(379,501)(380,502)(381,499)(382,500)(383,498)
(384,497);
s1 := Sym(512)!(  1,257)(  2,258)(  3,259)(  4,260)(  5,263)(  6,264)(  7,261)
(  8,262)(  9,265)( 10,266)( 11,267)( 12,268)( 13,271)( 14,272)( 15,269)
( 16,270)( 17,274)( 18,273)( 19,276)( 20,275)( 21,280)( 22,279)( 23,278)
( 24,277)( 25,282)( 26,281)( 27,284)( 28,283)( 29,288)( 30,287)( 31,286)
( 32,285)( 33,297)( 34,298)( 35,299)( 36,300)( 37,303)( 38,304)( 39,301)
( 40,302)( 41,289)( 42,290)( 43,291)( 44,292)( 45,295)( 46,296)( 47,293)
( 48,294)( 49,314)( 50,313)( 51,316)( 52,315)( 53,320)( 54,319)( 55,318)
( 56,317)( 57,306)( 58,305)( 59,308)( 60,307)( 61,312)( 62,311)( 63,310)
( 64,309)( 65,337)( 66,338)( 67,339)( 68,340)( 69,343)( 70,344)( 71,341)
( 72,342)( 73,345)( 74,346)( 75,347)( 76,348)( 77,351)( 78,352)( 79,349)
( 80,350)( 81,321)( 82,322)( 83,323)( 84,324)( 85,327)( 86,328)( 87,325)
( 88,326)( 89,329)( 90,330)( 91,331)( 92,332)( 93,335)( 94,336)( 95,333)
( 96,334)( 97,380)( 98,379)( 99,378)(100,377)(101,382)(102,381)(103,384)
(104,383)(105,372)(106,371)(107,370)(108,369)(109,374)(110,373)(111,376)
(112,375)(113,364)(114,363)(115,362)(116,361)(117,366)(118,365)(119,368)
(120,367)(121,356)(122,355)(123,354)(124,353)(125,358)(126,357)(127,360)
(128,359)(129,417)(130,418)(131,419)(132,420)(133,423)(134,424)(135,421)
(136,422)(137,425)(138,426)(139,427)(140,428)(141,431)(142,432)(143,429)
(144,430)(145,434)(146,433)(147,436)(148,435)(149,440)(150,439)(151,438)
(152,437)(153,442)(154,441)(155,444)(156,443)(157,448)(158,447)(159,446)
(160,445)(161,385)(162,386)(163,387)(164,388)(165,391)(166,392)(167,389)
(168,390)(169,393)(170,394)(171,395)(172,396)(173,399)(174,400)(175,397)
(176,398)(177,402)(178,401)(179,404)(180,403)(181,408)(182,407)(183,406)
(184,405)(185,410)(186,409)(187,412)(188,411)(189,416)(190,415)(191,414)
(192,413)(193,502)(194,501)(195,504)(196,503)(197,499)(198,500)(199,497)
(200,498)(201,510)(202,509)(203,512)(204,511)(205,507)(206,508)(207,505)
(208,506)(209,485)(210,486)(211,487)(212,488)(213,484)(214,483)(215,482)
(216,481)(217,493)(218,494)(219,495)(220,496)(221,492)(222,491)(223,490)
(224,489)(225,472)(226,471)(227,470)(228,469)(229,465)(230,466)(231,467)
(232,468)(233,480)(234,479)(235,478)(236,477)(237,473)(238,474)(239,475)
(240,476)(241,455)(242,456)(243,453)(244,454)(245,450)(246,449)(247,452)
(248,451)(249,463)(250,464)(251,461)(252,462)(253,458)(254,457)(255,460)
(256,459);
s2 := Sym(512)!(  1, 65)(  2, 66)(  3, 68)(  4, 67)(  5, 70)(  6, 69)(  7, 71)
(  8, 72)(  9, 73)( 10, 74)( 11, 76)( 12, 75)( 13, 78)( 14, 77)( 15, 79)
( 16, 80)( 17, 82)( 18, 81)( 19, 83)( 20, 84)( 21, 85)( 22, 86)( 23, 88)
( 24, 87)( 25, 90)( 26, 89)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 96)
( 32, 95)( 33,101)( 34,102)( 35,104)( 36,103)( 37, 97)( 38, 98)( 39,100)
( 40, 99)( 41,109)( 42,110)( 43,112)( 44,111)( 45,105)( 46,106)( 47,108)
( 48,107)( 49,117)( 50,118)( 51,120)( 52,119)( 53,113)( 54,114)( 55,116)
( 56,115)( 57,125)( 58,126)( 59,128)( 60,127)( 61,121)( 62,122)( 63,124)
( 64,123)(129,193)(130,194)(131,196)(132,195)(133,198)(134,197)(135,199)
(136,200)(137,201)(138,202)(139,204)(140,203)(141,206)(142,205)(143,207)
(144,208)(145,210)(146,209)(147,211)(148,212)(149,213)(150,214)(151,216)
(152,215)(153,218)(154,217)(155,219)(156,220)(157,221)(158,222)(159,224)
(160,223)(161,229)(162,230)(163,232)(164,231)(165,225)(166,226)(167,228)
(168,227)(169,237)(170,238)(171,240)(172,239)(173,233)(174,234)(175,236)
(176,235)(177,245)(178,246)(179,248)(180,247)(181,241)(182,242)(183,244)
(184,243)(185,253)(186,254)(187,256)(188,255)(189,249)(190,250)(191,252)
(192,251)(257,321)(258,322)(259,324)(260,323)(261,326)(262,325)(263,327)
(264,328)(265,329)(266,330)(267,332)(268,331)(269,334)(270,333)(271,335)
(272,336)(273,338)(274,337)(275,339)(276,340)(277,341)(278,342)(279,344)
(280,343)(281,346)(282,345)(283,347)(284,348)(285,349)(286,350)(287,352)
(288,351)(289,357)(290,358)(291,360)(292,359)(293,353)(294,354)(295,356)
(296,355)(297,365)(298,366)(299,368)(300,367)(301,361)(302,362)(303,364)
(304,363)(305,373)(306,374)(307,376)(308,375)(309,369)(310,370)(311,372)
(312,371)(313,381)(314,382)(315,384)(316,383)(317,377)(318,378)(319,380)
(320,379)(385,449)(386,450)(387,452)(388,451)(389,454)(390,453)(391,455)
(392,456)(393,457)(394,458)(395,460)(396,459)(397,462)(398,461)(399,463)
(400,464)(401,466)(402,465)(403,467)(404,468)(405,469)(406,470)(407,472)
(408,471)(409,474)(410,473)(411,475)(412,476)(413,477)(414,478)(415,480)
(416,479)(417,485)(418,486)(419,488)(420,487)(421,481)(422,482)(423,484)
(424,483)(425,493)(426,494)(427,496)(428,495)(429,489)(430,490)(431,492)
(432,491)(433,501)(434,502)(435,504)(436,503)(437,497)(438,498)(439,500)
(440,499)(441,509)(442,510)(443,512)(444,511)(445,505)(446,506)(447,508)
(448,507);
poly := sub<Sym(512)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s2 >; 
 
References : None.
to this polytope