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# Polytope of Type {16,4}

Atlas Canonical Name : {16,4}*512f
if this polytope has a name.
Group : SmallGroup(512,60821)
Rank : 3
Schlafli Type : {16,4}
Number of vertices, edges, etc : 64, 128, 16
Order of s0s1s2 : 8
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,4}*256b
4-fold quotients : {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)
(  9,204)( 10,203)( 11,202)( 12,201)( 13,207)( 14,208)( 15,205)( 16,206)
( 17,223)( 18,224)( 19,221)( 20,222)( 21,219)( 22,220)( 23,217)( 24,218)
( 25,215)( 26,216)( 27,213)( 28,214)( 29,211)( 30,212)( 31,209)( 32,210)
( 33,234)( 34,233)( 35,236)( 36,235)( 37,237)( 38,238)( 39,239)( 40,240)
( 41,226)( 42,225)( 43,228)( 44,227)( 45,229)( 46,230)( 47,231)( 48,232)
( 49,248)( 50,247)( 51,246)( 52,245)( 53,244)( 54,243)( 55,242)( 56,241)
( 57,253)( 58,254)( 59,255)( 60,256)( 61,249)( 62,250)( 63,251)( 64,252)
( 65,129)( 66,130)( 67,131)( 68,132)( 69,134)( 70,133)( 71,136)( 72,135)
( 73,140)( 74,139)( 75,138)( 76,137)( 77,143)( 78,144)( 79,141)( 80,142)
( 81,159)( 82,160)( 83,157)( 84,158)( 85,155)( 86,156)( 87,153)( 88,154)
( 89,151)( 90,152)( 91,149)( 92,150)( 93,147)( 94,148)( 95,145)( 96,146)
( 97,170)( 98,169)( 99,172)(100,171)(101,173)(102,174)(103,175)(104,176)
(105,162)(106,161)(107,164)(108,163)(109,165)(110,166)(111,167)(112,168)
(113,184)(114,183)(115,182)(116,181)(117,180)(118,179)(119,178)(120,177)
(121,189)(122,190)(123,191)(124,192)(125,185)(126,186)(127,187)(128,188)
(257,449)(258,450)(259,451)(260,452)(261,454)(262,453)(263,456)(264,455)
(265,460)(266,459)(267,458)(268,457)(269,463)(270,464)(271,461)(272,462)
(273,479)(274,480)(275,477)(276,478)(277,475)(278,476)(279,473)(280,474)
(281,471)(282,472)(283,469)(284,470)(285,467)(286,468)(287,465)(288,466)
(289,490)(290,489)(291,492)(292,491)(293,493)(294,494)(295,495)(296,496)
(297,482)(298,481)(299,484)(300,483)(301,485)(302,486)(303,487)(304,488)
(305,504)(306,503)(307,502)(308,501)(309,500)(310,499)(311,498)(312,497)
(313,509)(314,510)(315,511)(316,512)(317,505)(318,506)(319,507)(320,508)
(321,385)(322,386)(323,387)(324,388)(325,390)(326,389)(327,392)(328,391)
(329,396)(330,395)(331,394)(332,393)(333,399)(334,400)(335,397)(336,398)
(337,415)(338,416)(339,413)(340,414)(341,411)(342,412)(343,409)(344,410)
(345,407)(346,408)(347,405)(348,406)(349,403)(350,404)(351,401)(352,402)
(353,426)(354,425)(355,428)(356,427)(357,429)(358,430)(359,431)(360,432)
(361,418)(362,417)(363,420)(364,419)(365,421)(366,422)(367,423)(368,424)
(369,440)(370,439)(371,438)(372,437)(373,436)(374,435)(375,434)(376,433)
(377,445)(378,446)(379,447)(380,448)(381,441)(382,442)(383,443)(384,444);;
s1 := (  1,257)(  2,258)(  3,260)(  4,259)(  5,262)(  6,261)(  7,263)(  8,264)
(  9,269)( 10,270)( 11,272)( 12,271)( 13,265)( 14,266)( 15,268)( 16,267)
( 17,273)( 18,274)( 19,276)( 20,275)( 21,278)( 22,277)( 23,279)( 24,280)
( 25,285)( 26,286)( 27,288)( 28,287)( 29,281)( 30,282)( 31,284)( 32,283)
( 33,290)( 34,289)( 35,291)( 36,292)( 37,293)( 38,294)( 39,296)( 40,295)
( 41,302)( 42,301)( 43,303)( 44,304)( 45,298)( 46,297)( 47,299)( 48,300)
( 49,306)( 50,305)( 51,307)( 52,308)( 53,309)( 54,310)( 55,312)( 56,311)
( 57,318)( 58,317)( 59,319)( 60,320)( 61,314)( 62,313)( 63,315)( 64,316)
( 65,337)( 66,338)( 67,340)( 68,339)( 69,342)( 70,341)( 71,343)( 72,344)
( 73,349)( 74,350)( 75,352)( 76,351)( 77,345)( 78,346)( 79,348)( 80,347)
( 81,321)( 82,322)( 83,324)( 84,323)( 85,326)( 86,325)( 87,327)( 88,328)
( 89,333)( 90,334)( 91,336)( 92,335)( 93,329)( 94,330)( 95,332)( 96,331)
( 97,376)( 98,375)( 99,373)(100,374)(101,372)(102,371)(103,369)(104,370)
(105,379)(106,380)(107,378)(108,377)(109,384)(110,383)(111,381)(112,382)
(113,359)(114,360)(115,358)(116,357)(117,355)(118,356)(119,354)(120,353)
(121,364)(122,363)(123,361)(124,362)(125,367)(126,368)(127,366)(128,365)
(129,417)(130,418)(131,420)(132,419)(133,422)(134,421)(135,423)(136,424)
(137,429)(138,430)(139,432)(140,431)(141,425)(142,426)(143,428)(144,427)
(145,433)(146,434)(147,436)(148,435)(149,438)(150,437)(151,439)(152,440)
(153,445)(154,446)(155,448)(156,447)(157,441)(158,442)(159,444)(160,443)
(161,385)(162,386)(163,388)(164,387)(165,390)(166,389)(167,391)(168,392)
(169,397)(170,398)(171,400)(172,399)(173,393)(174,394)(175,396)(176,395)
(177,401)(178,402)(179,404)(180,403)(181,406)(182,405)(183,407)(184,408)
(185,413)(186,414)(187,416)(188,415)(189,409)(190,410)(191,412)(192,411)
(193,508)(194,507)(195,505)(196,506)(197,511)(198,512)(199,510)(200,509)
(201,501)(202,502)(203,504)(204,503)(205,497)(206,498)(207,500)(208,499)
(209,489)(210,490)(211,492)(212,491)(213,494)(214,493)(215,495)(216,496)
(217,488)(218,487)(219,485)(220,486)(221,484)(222,483)(223,481)(224,482)
(225,479)(226,480)(227,478)(228,477)(229,475)(230,476)(231,474)(232,473)
(233,465)(234,466)(235,468)(236,467)(237,470)(238,469)(239,471)(240,472)
(241,461)(242,462)(243,464)(244,463)(245,457)(246,458)(247,460)(248,459)
(249,451)(250,452)(251,450)(252,449)(253,456)(254,455)(255,453)(256,454);;
s2 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 72)(  6, 71)(  7, 70)(  8, 69)
(  9, 76)( 10, 75)( 11, 74)( 12, 73)( 13, 77)( 14, 78)( 15, 79)( 16, 80)
( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 88)( 22, 87)( 23, 86)( 24, 85)
( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 93)( 30, 94)( 31, 95)( 32, 96)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,112)( 38,111)( 39,110)( 40,109)
( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,104)( 46,103)( 47,102)( 48,101)
( 49,124)( 50,123)( 51,122)( 52,121)( 53,125)( 54,126)( 55,127)( 56,128)
( 57,116)( 58,115)( 59,114)( 60,113)( 61,117)( 62,118)( 63,119)( 64,120)
(129,193)(130,194)(131,195)(132,196)(133,200)(134,199)(135,198)(136,197)
(137,204)(138,203)(139,202)(140,201)(141,205)(142,206)(143,207)(144,208)
(145,209)(146,210)(147,211)(148,212)(149,216)(150,215)(151,214)(152,213)
(153,220)(154,219)(155,218)(156,217)(157,221)(158,222)(159,223)(160,224)
(161,233)(162,234)(163,235)(164,236)(165,240)(166,239)(167,238)(168,237)
(169,225)(170,226)(171,227)(172,228)(173,232)(174,231)(175,230)(176,229)
(177,252)(178,251)(179,250)(180,249)(181,253)(182,254)(183,255)(184,256)
(185,244)(186,243)(187,242)(188,241)(189,245)(190,246)(191,247)(192,248)
(257,321)(258,322)(259,323)(260,324)(261,328)(262,327)(263,326)(264,325)
(265,332)(266,331)(267,330)(268,329)(269,333)(270,334)(271,335)(272,336)
(273,337)(274,338)(275,339)(276,340)(277,344)(278,343)(279,342)(280,341)
(281,348)(282,347)(283,346)(284,345)(285,349)(286,350)(287,351)(288,352)
(289,361)(290,362)(291,363)(292,364)(293,368)(294,367)(295,366)(296,365)
(297,353)(298,354)(299,355)(300,356)(301,360)(302,359)(303,358)(304,357)
(305,380)(306,379)(307,378)(308,377)(309,381)(310,382)(311,383)(312,384)
(313,372)(314,371)(315,370)(316,369)(317,373)(318,374)(319,375)(320,376)
(385,449)(386,450)(387,451)(388,452)(389,456)(390,455)(391,454)(392,453)
(393,460)(394,459)(395,458)(396,457)(397,461)(398,462)(399,463)(400,464)
(401,465)(402,466)(403,467)(404,468)(405,472)(406,471)(407,470)(408,469)
(409,476)(410,475)(411,474)(412,473)(413,477)(414,478)(415,479)(416,480)
(417,489)(418,490)(419,491)(420,492)(421,496)(422,495)(423,494)(424,493)
(425,481)(426,482)(427,483)(428,484)(429,488)(430,487)(431,486)(432,485)
(433,508)(434,507)(435,506)(436,505)(437,509)(438,510)(439,511)(440,512)
(441,500)(442,499)(443,498)(444,497)(445,501)(446,502)(447,503)(448,504);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s0*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)(  9,204)( 10,203)( 11,202)( 12,201)( 13,207)( 14,208)( 15,205)
( 16,206)( 17,223)( 18,224)( 19,221)( 20,222)( 21,219)( 22,220)( 23,217)
( 24,218)( 25,215)( 26,216)( 27,213)( 28,214)( 29,211)( 30,212)( 31,209)
( 32,210)( 33,234)( 34,233)( 35,236)( 36,235)( 37,237)( 38,238)( 39,239)
( 40,240)( 41,226)( 42,225)( 43,228)( 44,227)( 45,229)( 46,230)( 47,231)
( 48,232)( 49,248)( 50,247)( 51,246)( 52,245)( 53,244)( 54,243)( 55,242)
( 56,241)( 57,253)( 58,254)( 59,255)( 60,256)( 61,249)( 62,250)( 63,251)
( 64,252)( 65,129)( 66,130)( 67,131)( 68,132)( 69,134)( 70,133)( 71,136)
( 72,135)( 73,140)( 74,139)( 75,138)( 76,137)( 77,143)( 78,144)( 79,141)
( 80,142)( 81,159)( 82,160)( 83,157)( 84,158)( 85,155)( 86,156)( 87,153)
( 88,154)( 89,151)( 90,152)( 91,149)( 92,150)( 93,147)( 94,148)( 95,145)
( 96,146)( 97,170)( 98,169)( 99,172)(100,171)(101,173)(102,174)(103,175)
(104,176)(105,162)(106,161)(107,164)(108,163)(109,165)(110,166)(111,167)
(112,168)(113,184)(114,183)(115,182)(116,181)(117,180)(118,179)(119,178)
(120,177)(121,189)(122,190)(123,191)(124,192)(125,185)(126,186)(127,187)
(128,188)(257,449)(258,450)(259,451)(260,452)(261,454)(262,453)(263,456)
(264,455)(265,460)(266,459)(267,458)(268,457)(269,463)(270,464)(271,461)
(272,462)(273,479)(274,480)(275,477)(276,478)(277,475)(278,476)(279,473)
(280,474)(281,471)(282,472)(283,469)(284,470)(285,467)(286,468)(287,465)
(288,466)(289,490)(290,489)(291,492)(292,491)(293,493)(294,494)(295,495)
(296,496)(297,482)(298,481)(299,484)(300,483)(301,485)(302,486)(303,487)
(304,488)(305,504)(306,503)(307,502)(308,501)(309,500)(310,499)(311,498)
(312,497)(313,509)(314,510)(315,511)(316,512)(317,505)(318,506)(319,507)
(320,508)(321,385)(322,386)(323,387)(324,388)(325,390)(326,389)(327,392)
(328,391)(329,396)(330,395)(331,394)(332,393)(333,399)(334,400)(335,397)
(336,398)(337,415)(338,416)(339,413)(340,414)(341,411)(342,412)(343,409)
(344,410)(345,407)(346,408)(347,405)(348,406)(349,403)(350,404)(351,401)
(352,402)(353,426)(354,425)(355,428)(356,427)(357,429)(358,430)(359,431)
(360,432)(361,418)(362,417)(363,420)(364,419)(365,421)(366,422)(367,423)
(368,424)(369,440)(370,439)(371,438)(372,437)(373,436)(374,435)(375,434)
(376,433)(377,445)(378,446)(379,447)(380,448)(381,441)(382,442)(383,443)
(384,444);
s1 := Sym(512)!(  1,257)(  2,258)(  3,260)(  4,259)(  5,262)(  6,261)(  7,263)
(  8,264)(  9,269)( 10,270)( 11,272)( 12,271)( 13,265)( 14,266)( 15,268)
( 16,267)( 17,273)( 18,274)( 19,276)( 20,275)( 21,278)( 22,277)( 23,279)
( 24,280)( 25,285)( 26,286)( 27,288)( 28,287)( 29,281)( 30,282)( 31,284)
( 32,283)( 33,290)( 34,289)( 35,291)( 36,292)( 37,293)( 38,294)( 39,296)
( 40,295)( 41,302)( 42,301)( 43,303)( 44,304)( 45,298)( 46,297)( 47,299)
( 48,300)( 49,306)( 50,305)( 51,307)( 52,308)( 53,309)( 54,310)( 55,312)
( 56,311)( 57,318)( 58,317)( 59,319)( 60,320)( 61,314)( 62,313)( 63,315)
( 64,316)( 65,337)( 66,338)( 67,340)( 68,339)( 69,342)( 70,341)( 71,343)
( 72,344)( 73,349)( 74,350)( 75,352)( 76,351)( 77,345)( 78,346)( 79,348)
( 80,347)( 81,321)( 82,322)( 83,324)( 84,323)( 85,326)( 86,325)( 87,327)
( 88,328)( 89,333)( 90,334)( 91,336)( 92,335)( 93,329)( 94,330)( 95,332)
( 96,331)( 97,376)( 98,375)( 99,373)(100,374)(101,372)(102,371)(103,369)
(104,370)(105,379)(106,380)(107,378)(108,377)(109,384)(110,383)(111,381)
(112,382)(113,359)(114,360)(115,358)(116,357)(117,355)(118,356)(119,354)
(120,353)(121,364)(122,363)(123,361)(124,362)(125,367)(126,368)(127,366)
(128,365)(129,417)(130,418)(131,420)(132,419)(133,422)(134,421)(135,423)
(136,424)(137,429)(138,430)(139,432)(140,431)(141,425)(142,426)(143,428)
(144,427)(145,433)(146,434)(147,436)(148,435)(149,438)(150,437)(151,439)
(152,440)(153,445)(154,446)(155,448)(156,447)(157,441)(158,442)(159,444)
(160,443)(161,385)(162,386)(163,388)(164,387)(165,390)(166,389)(167,391)
(168,392)(169,397)(170,398)(171,400)(172,399)(173,393)(174,394)(175,396)
(176,395)(177,401)(178,402)(179,404)(180,403)(181,406)(182,405)(183,407)
(184,408)(185,413)(186,414)(187,416)(188,415)(189,409)(190,410)(191,412)
(192,411)(193,508)(194,507)(195,505)(196,506)(197,511)(198,512)(199,510)
(200,509)(201,501)(202,502)(203,504)(204,503)(205,497)(206,498)(207,500)
(208,499)(209,489)(210,490)(211,492)(212,491)(213,494)(214,493)(215,495)
(216,496)(217,488)(218,487)(219,485)(220,486)(221,484)(222,483)(223,481)
(224,482)(225,479)(226,480)(227,478)(228,477)(229,475)(230,476)(231,474)
(232,473)(233,465)(234,466)(235,468)(236,467)(237,470)(238,469)(239,471)
(240,472)(241,461)(242,462)(243,464)(244,463)(245,457)(246,458)(247,460)
(248,459)(249,451)(250,452)(251,450)(252,449)(253,456)(254,455)(255,453)
(256,454);
s2 := Sym(512)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 72)(  6, 71)(  7, 70)
(  8, 69)(  9, 76)( 10, 75)( 11, 74)( 12, 73)( 13, 77)( 14, 78)( 15, 79)
( 16, 80)( 17, 81)( 18, 82)( 19, 83)( 20, 84)( 21, 88)( 22, 87)( 23, 86)
( 24, 85)( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 93)( 30, 94)( 31, 95)
( 32, 96)( 33,105)( 34,106)( 35,107)( 36,108)( 37,112)( 38,111)( 39,110)
( 40,109)( 41, 97)( 42, 98)( 43, 99)( 44,100)( 45,104)( 46,103)( 47,102)
( 48,101)( 49,124)( 50,123)( 51,122)( 52,121)( 53,125)( 54,126)( 55,127)
( 56,128)( 57,116)( 58,115)( 59,114)( 60,113)( 61,117)( 62,118)( 63,119)
( 64,120)(129,193)(130,194)(131,195)(132,196)(133,200)(134,199)(135,198)
(136,197)(137,204)(138,203)(139,202)(140,201)(141,205)(142,206)(143,207)
(144,208)(145,209)(146,210)(147,211)(148,212)(149,216)(150,215)(151,214)
(152,213)(153,220)(154,219)(155,218)(156,217)(157,221)(158,222)(159,223)
(160,224)(161,233)(162,234)(163,235)(164,236)(165,240)(166,239)(167,238)
(168,237)(169,225)(170,226)(171,227)(172,228)(173,232)(174,231)(175,230)
(176,229)(177,252)(178,251)(179,250)(180,249)(181,253)(182,254)(183,255)
(184,256)(185,244)(186,243)(187,242)(188,241)(189,245)(190,246)(191,247)
(192,248)(257,321)(258,322)(259,323)(260,324)(261,328)(262,327)(263,326)
(264,325)(265,332)(266,331)(267,330)(268,329)(269,333)(270,334)(271,335)
(272,336)(273,337)(274,338)(275,339)(276,340)(277,344)(278,343)(279,342)
(280,341)(281,348)(282,347)(283,346)(284,345)(285,349)(286,350)(287,351)
(288,352)(289,361)(290,362)(291,363)(292,364)(293,368)(294,367)(295,366)
(296,365)(297,353)(298,354)(299,355)(300,356)(301,360)(302,359)(303,358)
(304,357)(305,380)(306,379)(307,378)(308,377)(309,381)(310,382)(311,383)
(312,384)(313,372)(314,371)(315,370)(316,369)(317,373)(318,374)(319,375)
(320,376)(385,449)(386,450)(387,451)(388,452)(389,456)(390,455)(391,454)
(392,453)(393,460)(394,459)(395,458)(396,457)(397,461)(398,462)(399,463)
(400,464)(401,465)(402,466)(403,467)(404,468)(405,472)(406,471)(407,470)
(408,469)(409,476)(410,475)(411,474)(412,473)(413,477)(414,478)(415,479)
(416,480)(417,489)(418,490)(419,491)(420,492)(421,496)(422,495)(423,494)
(424,493)(425,481)(426,482)(427,483)(428,484)(429,488)(430,487)(431,486)
(432,485)(433,508)(434,507)(435,506)(436,505)(437,509)(438,510)(439,511)
(440,512)(441,500)(442,499)(443,498)(444,497)(445,501)(446,502)(447,503)
(448,504);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1 >;

```
References : None.
to this polytope