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# Polytope of Type {8,8}

Atlas Canonical Name : {8,8}*512i
if this polytope has a name.
Group : SmallGroup(512,58338)
Rank : 3
Schlafli Type : {8,8}
Number of vertices, edges, etc : 32, 128, 32
Order of s0s1s2 : 4
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 : {8,8}*256b, {8,4}*256b, {8,8}*256f
4-fold quotients : {4,8}*128b, {8,4}*128b
8-fold quotients : {4,4}*64
16-fold quotients : {4,4}*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 := (  1,193)(  2,194)(  3,195)(  4,196)(  5,198)(  6,197)(  7,200)(  8,199)
(  9,201)( 10,202)( 11,203)( 12,204)( 13,206)( 14,205)( 15,208)( 16,207)
( 17,216)( 18,215)( 19,214)( 20,213)( 21,212)( 22,211)( 23,210)( 24,209)
( 25,224)( 26,223)( 27,222)( 28,221)( 29,220)( 30,219)( 31,218)( 32,217)
( 33,237)( 34,238)( 35,239)( 36,240)( 37,233)( 38,234)( 39,235)( 40,236)
( 41,229)( 42,230)( 43,231)( 44,232)( 45,225)( 46,226)( 47,227)( 48,228)
( 49,252)( 50,251)( 51,250)( 52,249)( 53,255)( 54,256)( 55,253)( 56,254)
( 57,244)( 58,243)( 59,242)( 60,241)( 61,247)( 62,248)( 63,245)( 64,246)
( 65,129)( 66,130)( 67,131)( 68,132)( 69,134)( 70,133)( 71,136)( 72,135)
( 73,137)( 74,138)( 75,139)( 76,140)( 77,142)( 78,141)( 79,144)( 80,143)
( 81,152)( 82,151)( 83,150)( 84,149)( 85,148)( 86,147)( 87,146)( 88,145)
( 89,160)( 90,159)( 91,158)( 92,157)( 93,156)( 94,155)( 95,154)( 96,153)
( 97,173)( 98,174)( 99,175)(100,176)(101,169)(102,170)(103,171)(104,172)
(105,165)(106,166)(107,167)(108,168)(109,161)(110,162)(111,163)(112,164)
(113,188)(114,187)(115,186)(116,185)(117,191)(118,192)(119,189)(120,190)
(121,180)(122,179)(123,178)(124,177)(125,183)(126,184)(127,181)(128,182)
(257,449)(258,450)(259,451)(260,452)(261,454)(262,453)(263,456)(264,455)
(265,457)(266,458)(267,459)(268,460)(269,462)(270,461)(271,464)(272,463)
(273,472)(274,471)(275,470)(276,469)(277,468)(278,467)(279,466)(280,465)
(281,480)(282,479)(283,478)(284,477)(285,476)(286,475)(287,474)(288,473)
(289,493)(290,494)(291,495)(292,496)(293,489)(294,490)(295,491)(296,492)
(297,485)(298,486)(299,487)(300,488)(301,481)(302,482)(303,483)(304,484)
(305,508)(306,507)(307,506)(308,505)(309,511)(310,512)(311,509)(312,510)
(313,500)(314,499)(315,498)(316,497)(317,503)(318,504)(319,501)(320,502)
(321,385)(322,386)(323,387)(324,388)(325,390)(326,389)(327,392)(328,391)
(329,393)(330,394)(331,395)(332,396)(333,398)(334,397)(335,400)(336,399)
(337,408)(338,407)(339,406)(340,405)(341,404)(342,403)(343,402)(344,401)
(345,416)(346,415)(347,414)(348,413)(349,412)(350,411)(351,410)(352,409)
(353,429)(354,430)(355,431)(356,432)(357,425)(358,426)(359,427)(360,428)
(361,421)(362,422)(363,423)(364,424)(365,417)(366,418)(367,419)(368,420)
(369,444)(370,443)(371,442)(372,441)(373,447)(374,448)(375,445)(376,446)
(377,436)(378,435)(379,434)(380,433)(381,439)(382,440)(383,437)(384,438);;
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)
( 17,273)( 18,274)( 19,275)( 20,276)( 21,279)( 22,280)( 23,277)( 24,278)
( 25,281)( 26,282)( 27,283)( 28,284)( 29,287)( 30,288)( 31,285)( 32,286)
( 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)
( 49,313)( 50,314)( 51,315)( 52,316)( 53,319)( 54,320)( 55,317)( 56,318)
( 57,305)( 58,306)( 59,307)( 60,308)( 61,311)( 62,312)( 63,309)( 64,310)
( 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)
(145,433)(146,434)(147,435)(148,436)(149,439)(150,440)(151,437)(152,438)
(153,441)(154,442)(155,443)(156,444)(157,447)(158,448)(159,445)(160,446)
(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)
(177,401)(178,402)(179,403)(180,404)(181,407)(182,408)(183,405)(184,406)
(185,409)(186,410)(187,411)(188,412)(189,415)(190,416)(191,413)(192,414)
(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 := (  1,129)(  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);;
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(512)!(  1,193)(  2,194)(  3,195)(  4,196)(  5,198)(  6,197)(  7,200)
(  8,199)(  9,201)( 10,202)( 11,203)( 12,204)( 13,206)( 14,205)( 15,208)
( 16,207)( 17,216)( 18,215)( 19,214)( 20,213)( 21,212)( 22,211)( 23,210)
( 24,209)( 25,224)( 26,223)( 27,222)( 28,221)( 29,220)( 30,219)( 31,218)
( 32,217)( 33,237)( 34,238)( 35,239)( 36,240)( 37,233)( 38,234)( 39,235)
( 40,236)( 41,229)( 42,230)( 43,231)( 44,232)( 45,225)( 46,226)( 47,227)
( 48,228)( 49,252)( 50,251)( 51,250)( 52,249)( 53,255)( 54,256)( 55,253)
( 56,254)( 57,244)( 58,243)( 59,242)( 60,241)( 61,247)( 62,248)( 63,245)
( 64,246)( 65,129)( 66,130)( 67,131)( 68,132)( 69,134)( 70,133)( 71,136)
( 72,135)( 73,137)( 74,138)( 75,139)( 76,140)( 77,142)( 78,141)( 79,144)
( 80,143)( 81,152)( 82,151)( 83,150)( 84,149)( 85,148)( 86,147)( 87,146)
( 88,145)( 89,160)( 90,159)( 91,158)( 92,157)( 93,156)( 94,155)( 95,154)
( 96,153)( 97,173)( 98,174)( 99,175)(100,176)(101,169)(102,170)(103,171)
(104,172)(105,165)(106,166)(107,167)(108,168)(109,161)(110,162)(111,163)
(112,164)(113,188)(114,187)(115,186)(116,185)(117,191)(118,192)(119,189)
(120,190)(121,180)(122,179)(123,178)(124,177)(125,183)(126,184)(127,181)
(128,182)(257,449)(258,450)(259,451)(260,452)(261,454)(262,453)(263,456)
(264,455)(265,457)(266,458)(267,459)(268,460)(269,462)(270,461)(271,464)
(272,463)(273,472)(274,471)(275,470)(276,469)(277,468)(278,467)(279,466)
(280,465)(281,480)(282,479)(283,478)(284,477)(285,476)(286,475)(287,474)
(288,473)(289,493)(290,494)(291,495)(292,496)(293,489)(294,490)(295,491)
(296,492)(297,485)(298,486)(299,487)(300,488)(301,481)(302,482)(303,483)
(304,484)(305,508)(306,507)(307,506)(308,505)(309,511)(310,512)(311,509)
(312,510)(313,500)(314,499)(315,498)(316,497)(317,503)(318,504)(319,501)
(320,502)(321,385)(322,386)(323,387)(324,388)(325,390)(326,389)(327,392)
(328,391)(329,393)(330,394)(331,395)(332,396)(333,398)(334,397)(335,400)
(336,399)(337,408)(338,407)(339,406)(340,405)(341,404)(342,403)(343,402)
(344,401)(345,416)(346,415)(347,414)(348,413)(349,412)(350,411)(351,410)
(352,409)(353,429)(354,430)(355,431)(356,432)(357,425)(358,426)(359,427)
(360,428)(361,421)(362,422)(363,423)(364,424)(365,417)(366,418)(367,419)
(368,420)(369,444)(370,443)(371,442)(372,441)(373,447)(374,448)(375,445)
(376,446)(377,436)(378,435)(379,434)(380,433)(381,439)(382,440)(383,437)
(384,438);
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)( 17,273)( 18,274)( 19,275)( 20,276)( 21,279)( 22,280)( 23,277)
( 24,278)( 25,281)( 26,282)( 27,283)( 28,284)( 29,287)( 30,288)( 31,285)
( 32,286)( 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)( 49,313)( 50,314)( 51,315)( 52,316)( 53,319)( 54,320)( 55,317)
( 56,318)( 57,305)( 58,306)( 59,307)( 60,308)( 61,311)( 62,312)( 63,309)
( 64,310)( 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)(145,433)(146,434)(147,435)(148,436)(149,439)(150,440)(151,437)
(152,438)(153,441)(154,442)(155,443)(156,444)(157,447)(158,448)(159,445)
(160,446)(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)(177,401)(178,402)(179,403)(180,404)(181,407)(182,408)(183,405)
(184,406)(185,409)(186,410)(187,411)(188,412)(189,415)(190,416)(191,413)
(192,414)(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)!(  1,129)(  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);
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 >;

```
References : None.
to this polytope