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# Polytope of Type {8,16}

Atlas Canonical Name : {8,16}*512a
if this polytope has a name.
Group : SmallGroup(512,30247)
Rank : 3
Schlafli Type : {8,16}
Number of vertices, edges, etc : 16, 128, 32
Order of s0s1s2 : 16
Order of s0s1s2s1 : 4
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, {4,16}*256a
4-fold quotients : {4,8}*128a, {4,16}*128a, {4,16}*128b, {8,4}*128b
8-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64, {2,16}*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, 69)(  6, 70)(  7, 71)(  8, 72)
(  9, 73)( 10, 74)( 11, 75)( 12, 76)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 82)( 18, 81)( 19, 84)( 20, 83)( 21, 86)( 22, 85)( 23, 88)( 24, 87)
( 25, 90)( 26, 89)( 27, 92)( 28, 91)( 29, 94)( 30, 93)( 31, 96)( 32, 95)
( 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)
( 49,118)( 50,117)( 51,120)( 52,119)( 53,114)( 54,113)( 55,116)( 56,115)
( 57,126)( 58,125)( 59,128)( 60,127)( 61,122)( 62,121)( 63,124)( 64,123)
(129,193)(130,194)(131,195)(132,196)(133,197)(134,198)(135,199)(136,200)
(137,201)(138,202)(139,203)(140,204)(141,205)(142,206)(143,207)(144,208)
(145,210)(146,209)(147,212)(148,211)(149,214)(150,213)(151,216)(152,215)
(153,218)(154,217)(155,220)(156,219)(157,222)(158,221)(159,224)(160,223)
(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)
(177,246)(178,245)(179,248)(180,247)(181,242)(182,241)(183,244)(184,243)
(185,254)(186,253)(187,256)(188,255)(189,250)(190,249)(191,252)(192,251)
(257,321)(258,322)(259,323)(260,324)(261,325)(262,326)(263,327)(264,328)
(265,329)(266,330)(267,331)(268,332)(269,333)(270,334)(271,335)(272,336)
(273,338)(274,337)(275,340)(276,339)(277,342)(278,341)(279,344)(280,343)
(281,346)(282,345)(283,348)(284,347)(285,350)(286,349)(287,352)(288,351)
(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)
(305,374)(306,373)(307,376)(308,375)(309,370)(310,369)(311,372)(312,371)
(313,382)(314,381)(315,384)(316,383)(317,378)(318,377)(319,380)(320,379)
(385,449)(386,450)(387,451)(388,452)(389,453)(390,454)(391,455)(392,456)
(393,457)(394,458)(395,459)(396,460)(397,461)(398,462)(399,463)(400,464)
(401,466)(402,465)(403,468)(404,467)(405,470)(406,469)(407,472)(408,471)
(409,474)(410,473)(411,476)(412,475)(413,478)(414,477)(415,480)(416,479)
(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)
(433,502)(434,501)(435,504)(436,503)(437,498)(438,497)(439,500)(440,499)
(441,510)(442,509)(443,512)(444,511)(445,506)(446,505)(447,508)(448,507);;
s1 := (  1,257)(  2,258)(  3,259)(  4,260)(  5,262)(  6,261)(  7,264)(  8,263)
(  9,267)( 10,268)( 11,265)( 12,266)( 13,272)( 14,271)( 15,270)( 16,269)
( 17,274)( 18,273)( 19,276)( 20,275)( 21,277)( 22,278)( 23,279)( 24,280)
( 25,284)( 26,283)( 27,282)( 28,281)( 29,287)( 30,288)( 31,285)( 32,286)
( 33,297)( 34,298)( 35,299)( 36,300)( 37,302)( 38,301)( 39,304)( 40,303)
( 41,289)( 42,290)( 43,291)( 44,292)( 45,294)( 46,293)( 47,296)( 48,295)
( 49,314)( 50,313)( 51,316)( 52,315)( 53,317)( 54,318)( 55,319)( 56,320)
( 57,306)( 58,305)( 59,308)( 60,307)( 61,309)( 62,310)( 63,311)( 64,312)
( 65,337)( 66,338)( 67,339)( 68,340)( 69,342)( 70,341)( 71,344)( 72,343)
( 73,347)( 74,348)( 75,345)( 76,346)( 77,352)( 78,351)( 79,350)( 80,349)
( 81,321)( 82,322)( 83,323)( 84,324)( 85,326)( 86,325)( 87,328)( 88,327)
( 89,331)( 90,332)( 91,329)( 92,330)( 93,336)( 94,335)( 95,334)( 96,333)
( 97,378)( 98,377)( 99,380)(100,379)(101,381)(102,382)(103,383)(104,384)
(105,370)(106,369)(107,372)(108,371)(109,373)(110,374)(111,375)(112,376)
(113,362)(114,361)(115,364)(116,363)(117,365)(118,366)(119,367)(120,368)
(121,354)(122,353)(123,356)(124,355)(125,357)(126,358)(127,359)(128,360)
(129,417)(130,418)(131,419)(132,420)(133,422)(134,421)(135,424)(136,423)
(137,427)(138,428)(139,425)(140,426)(141,432)(142,431)(143,430)(144,429)
(145,434)(146,433)(147,436)(148,435)(149,437)(150,438)(151,439)(152,440)
(153,444)(154,443)(155,442)(156,441)(157,447)(158,448)(159,445)(160,446)
(161,385)(162,386)(163,387)(164,388)(165,390)(166,389)(167,392)(168,391)
(169,395)(170,396)(171,393)(172,394)(173,400)(174,399)(175,398)(176,397)
(177,402)(178,401)(179,404)(180,403)(181,405)(182,406)(183,407)(184,408)
(185,412)(186,411)(187,410)(188,409)(189,415)(190,416)(191,413)(192,414)
(193,501)(194,502)(195,503)(196,504)(197,498)(198,497)(199,500)(200,499)
(201,511)(202,512)(203,509)(204,510)(205,508)(206,507)(207,506)(208,505)
(209,485)(210,486)(211,487)(212,488)(213,482)(214,481)(215,484)(216,483)
(217,495)(218,496)(219,493)(220,494)(221,492)(222,491)(223,490)(224,489)
(225,470)(226,469)(227,472)(228,471)(229,465)(230,466)(231,467)(232,468)
(233,480)(234,479)(235,478)(236,477)(237,475)(238,476)(239,473)(240,474)
(241,454)(242,453)(243,456)(244,455)(245,449)(246,450)(247,451)(248,452)
(249,464)(250,463)(251,462)(252,461)(253,459)(254,460)(255,457)(256,458);;
s2 := (  1,129)(  2,130)(  3,131)(  4,132)(  5,133)(  6,134)(  7,135)(  8,136)
(  9,139)( 10,140)( 11,137)( 12,138)( 13,143)( 14,144)( 15,141)( 16,142)
( 17,150)( 18,149)( 19,152)( 20,151)( 21,146)( 22,145)( 23,148)( 24,147)
( 25,160)( 26,159)( 27,158)( 28,157)( 29,156)( 30,155)( 31,154)( 32,153)
( 33,169)( 34,170)( 35,171)( 36,172)( 37,173)( 38,174)( 39,175)( 40,176)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,190)( 50,189)( 51,192)( 52,191)( 53,186)( 54,185)( 55,188)( 56,187)
( 57,182)( 58,181)( 59,184)( 60,183)( 61,178)( 62,177)( 63,180)( 64,179)
( 65,193)( 66,194)( 67,195)( 68,196)( 69,197)( 70,198)( 71,199)( 72,200)
( 73,203)( 74,204)( 75,201)( 76,202)( 77,207)( 78,208)( 79,205)( 80,206)
( 81,214)( 82,213)( 83,216)( 84,215)( 85,210)( 86,209)( 87,212)( 88,211)
( 89,224)( 90,223)( 91,222)( 92,221)( 93,220)( 94,219)( 95,218)( 96,217)
( 97,233)( 98,234)( 99,235)(100,236)(101,237)(102,238)(103,239)(104,240)
(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)
(113,254)(114,253)(115,256)(116,255)(117,250)(118,249)(119,252)(120,251)
(121,246)(122,245)(123,248)(124,247)(125,242)(126,241)(127,244)(128,243)
(257,385)(258,386)(259,387)(260,388)(261,389)(262,390)(263,391)(264,392)
(265,395)(266,396)(267,393)(268,394)(269,399)(270,400)(271,397)(272,398)
(273,406)(274,405)(275,408)(276,407)(277,402)(278,401)(279,404)(280,403)
(281,416)(282,415)(283,414)(284,413)(285,412)(286,411)(287,410)(288,409)
(289,425)(290,426)(291,427)(292,428)(293,429)(294,430)(295,431)(296,432)
(297,417)(298,418)(299,419)(300,420)(301,421)(302,422)(303,423)(304,424)
(305,446)(306,445)(307,448)(308,447)(309,442)(310,441)(311,444)(312,443)
(313,438)(314,437)(315,440)(316,439)(317,434)(318,433)(319,436)(320,435)
(321,449)(322,450)(323,451)(324,452)(325,453)(326,454)(327,455)(328,456)
(329,459)(330,460)(331,457)(332,458)(333,463)(334,464)(335,461)(336,462)
(337,470)(338,469)(339,472)(340,471)(341,466)(342,465)(343,468)(344,467)
(345,480)(346,479)(347,478)(348,477)(349,476)(350,475)(351,474)(352,473)
(353,489)(354,490)(355,491)(356,492)(357,493)(358,494)(359,495)(360,496)
(361,481)(362,482)(363,483)(364,484)(365,485)(366,486)(367,487)(368,488)
(369,510)(370,509)(371,512)(372,511)(373,506)(374,505)(375,508)(376,507)
(377,502)(378,501)(379,504)(380,503)(381,498)(382,497)(383,500)(384,499);;
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*s0*s1*s2*s1*s2*s1*s0*s1,
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

References : None.
to this polytope