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# Polytope of Type {8,16}

Atlas Canonical Name : {8,16}*512h
if this polytope has a name.
Group : SmallGroup(512,58364)
Rank : 3
Schlafli Type : {8,16}
Number of vertices, edges, etc : 16, 128, 32
Order of s0s1s2 : 16
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,8}*256c
4-fold quotients : {4,8}*128a, {8,4}*128b
8-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64
16-fold quotients : {4,4}*32, {2,8}*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, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 70)(  6, 69)(  7, 72)(  8, 71)
(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 78)( 14, 77)( 15, 80)( 16, 79)
( 17, 84)( 18, 83)( 19, 82)( 20, 81)( 21, 87)( 22, 88)( 23, 85)( 24, 86)
( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 95)( 30, 96)( 31, 93)( 32, 94)
( 33,101)( 34,102)( 35,103)( 36,104)( 37, 97)( 38, 98)( 39, 99)( 40,100)
( 41,109)( 42,110)( 43,111)( 44,112)( 45,105)( 46,106)( 47,107)( 48,108)
( 49,120)( 50,119)( 51,118)( 52,117)( 53,116)( 54,115)( 55,114)( 56,113)
( 57,128)( 58,127)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)( 64,121)
(129,193)(130,194)(131,195)(132,196)(133,198)(134,197)(135,200)(136,199)
(137,201)(138,202)(139,203)(140,204)(141,206)(142,205)(143,208)(144,207)
(145,212)(146,211)(147,210)(148,209)(149,215)(150,216)(151,213)(152,214)
(153,220)(154,219)(155,218)(156,217)(157,223)(158,224)(159,221)(160,222)
(161,229)(162,230)(163,231)(164,232)(165,225)(166,226)(167,227)(168,228)
(169,237)(170,238)(171,239)(172,240)(173,233)(174,234)(175,235)(176,236)
(177,248)(178,247)(179,246)(180,245)(181,244)(182,243)(183,242)(184,241)
(185,256)(186,255)(187,254)(188,253)(189,252)(190,251)(191,250)(192,249)
(257,321)(258,322)(259,323)(260,324)(261,326)(262,325)(263,328)(264,327)
(265,329)(266,330)(267,331)(268,332)(269,334)(270,333)(271,336)(272,335)
(273,340)(274,339)(275,338)(276,337)(277,343)(278,344)(279,341)(280,342)
(281,348)(282,347)(283,346)(284,345)(285,351)(286,352)(287,349)(288,350)
(289,357)(290,358)(291,359)(292,360)(293,353)(294,354)(295,355)(296,356)
(297,365)(298,366)(299,367)(300,368)(301,361)(302,362)(303,363)(304,364)
(305,376)(306,375)(307,374)(308,373)(309,372)(310,371)(311,370)(312,369)
(313,384)(314,383)(315,382)(316,381)(317,380)(318,379)(319,378)(320,377)
(385,449)(386,450)(387,451)(388,452)(389,454)(390,453)(391,456)(392,455)
(393,457)(394,458)(395,459)(396,460)(397,462)(398,461)(399,464)(400,463)
(401,468)(402,467)(403,466)(404,465)(405,471)(406,472)(407,469)(408,470)
(409,476)(410,475)(411,474)(412,473)(413,479)(414,480)(415,477)(416,478)
(417,485)(418,486)(419,487)(420,488)(421,481)(422,482)(423,483)(424,484)
(425,493)(426,494)(427,495)(428,496)(429,489)(430,490)(431,491)(432,492)
(433,504)(434,503)(435,502)(436,501)(437,500)(438,499)(439,498)(440,497)
(441,512)(442,511)(443,510)(444,509)(445,508)(446,507)(447,506)(448,505);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,263)(  6,264)(  7,261)(  8,262)
(  9,266)( 10,265)( 11,268)( 12,267)( 13,272)( 14,271)( 15,270)( 16,269)
( 17,276)( 18,275)( 19,274)( 20,273)( 21,278)( 22,277)( 23,280)( 24,279)
( 25,283)( 26,284)( 27,281)( 28,282)( 29,285)( 30,286)( 31,287)( 32,288)
( 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)
( 49,316)( 50,315)( 51,314)( 52,313)( 53,318)( 54,317)( 55,320)( 56,319)
( 57,308)( 58,307)( 59,306)( 60,305)( 61,310)( 62,309)( 63,312)( 64,311)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,343)( 70,344)( 71,341)( 72,342)
( 73,346)( 74,345)( 75,348)( 76,347)( 77,352)( 78,351)( 79,350)( 80,349)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,327)( 86,328)( 87,325)( 88,326)
( 89,330)( 90,329)( 91,332)( 92,331)( 93,336)( 94,335)( 95,334)( 96,333)
( 97,379)( 98,380)( 99,377)(100,378)(101,381)(102,382)(103,383)(104,384)
(105,371)(106,372)(107,369)(108,370)(109,373)(110,374)(111,375)(112,376)
(113,363)(114,364)(115,361)(116,362)(117,365)(118,366)(119,367)(120,368)
(121,355)(122,356)(123,353)(124,354)(125,357)(126,358)(127,359)(128,360)
(129,417)(130,418)(131,419)(132,420)(133,423)(134,424)(135,421)(136,422)
(137,426)(138,425)(139,428)(140,427)(141,432)(142,431)(143,430)(144,429)
(145,436)(146,435)(147,434)(148,433)(149,438)(150,437)(151,440)(152,439)
(153,443)(154,444)(155,441)(156,442)(157,445)(158,446)(159,447)(160,448)
(161,385)(162,386)(163,387)(164,388)(165,391)(166,392)(167,389)(168,390)
(169,394)(170,393)(171,396)(172,395)(173,400)(174,399)(175,398)(176,397)
(177,404)(178,403)(179,402)(180,401)(181,406)(182,405)(183,408)(184,407)
(185,411)(186,412)(187,409)(188,410)(189,413)(190,414)(191,415)(192,416)
(193,501)(194,502)(195,503)(196,504)(197,500)(198,499)(199,498)(200,497)
(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)
(217,494)(218,493)(219,496)(220,495)(221,491)(222,492)(223,489)(224,490)
(225,472)(226,471)(227,470)(228,469)(229,465)(230,466)(231,467)(232,468)
(233,479)(234,480)(235,477)(236,478)(237,474)(238,473)(239,476)(240,475)
(241,456)(242,455)(243,454)(244,453)(245,449)(246,450)(247,451)(248,452)
(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)
(  9,138)( 10,137)( 11,139)( 12,140)( 13,142)( 14,141)( 15,143)( 16,144)
( 17,152)( 18,151)( 19,149)( 20,150)( 21,147)( 22,148)( 23,146)( 24,145)
( 25,159)( 26,160)( 27,158)( 28,157)( 29,156)( 30,155)( 31,153)( 32,154)
( 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)
( 73,202)( 74,201)( 75,203)( 76,204)( 77,206)( 78,205)( 79,207)( 80,208)
( 81,216)( 82,215)( 83,213)( 84,214)( 85,211)( 86,212)( 87,210)( 88,209)
( 89,223)( 90,224)( 91,222)( 92,221)( 93,220)( 94,219)( 95,217)( 96,218)
( 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)
(265,394)(266,393)(267,395)(268,396)(269,398)(270,397)(271,399)(272,400)
(273,408)(274,407)(275,405)(276,406)(277,403)(278,404)(279,402)(280,401)
(281,415)(282,416)(283,414)(284,413)(285,412)(286,411)(287,409)(288,410)
(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)
(329,458)(330,457)(331,459)(332,460)(333,462)(334,461)(335,463)(336,464)
(337,472)(338,471)(339,469)(340,470)(341,467)(342,468)(343,466)(344,465)
(345,479)(346,480)(347,478)(348,477)(349,476)(350,475)(351,473)(352,474)
(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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0 ];;
poly := F / rels;;

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

```
References : None.
to this polytope