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Dic₁₄₂₅₂ #72290785

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

Splittings

Subgroups

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 532 points:

(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)(18,23)(19,22)(20,21)(25,532)(26,531)(27,530)(28,529)(29,528)(30,527)(31,526)(32,525)(33,524)(34,523)(35,522)(36,521)(37,520)(38,519)(39,518)(40,517)(41,516)(42,515)(43,514)(44,513)(45,512)(46,511)(47,510)(48,509)(49,508)(50,507)(51,506)(52,505)(53,504)(54,503)(55,502)(56,501)(57,500)(58,499)(59,498)(60,497)(61,496)(62,495)(63,494)(64,493)(65,492)(66,491)(67,490)(68,489)(69,488)(70,487)(71,486)(72,485)(73,484)(74,483)(75,482)(76,481)(77,480)(78,479)(79,478)(80,477)(81,476)(82,475)(83,474)(84,473)(85,472)(86,471)(87,470)(88,469)(89,468)(90,467)(91,466)(92,465)(93,464)(94,463)(95,462)(96,461)(97,460)(98,459)(99,458)(100,457)(101,456)(102,455)(103,454)(104,453)(105,452)(106,451)(107,450)(108,449)(109,448)(110,447)(111,446)(112,445)(113,444)(114,443)(115,442)(116,441)(117,440)(118,439)(119,438)(120,437)(121,436)(122,435)(123,434)(124,433)(125,432)(126,431)(127,430)(128,429)(129,428)(130,427)(131,426)(132,425)(133,424)(134,423)(135,422)(136,421)(137,420)(138,419)(139,418)(140,417)(141,416)(142,415)(143,414)(144,413)(145,412)(146,411)(147,410)(148,409)(149,408)(150,407)(151,406)(152,405)(153,404)(154,403)(155,402)(156,401)(157,400)(158,399)(159,398)(160,397)(161,396)(162,395)(163,394)(164,393)(165,392)(166,391)(167,390)(168,389)(169,388)(170,387)(171,386)(172,385)(173,384)(174,383)(175,382)(176,381)(177,380)(178,379)(179,378)(180,377)(181,376)(182,375)(183,374)(184,373)(185,372)(186,371)(187,370)(188,369)(189,368)(190,367)(191,366)(192,365)(193,364)(194,363)(195,362)(196,361)(197,360)(198,359)(199,358)(200,357)(201,356)(202,355)(203,354)(204,353)(205,352)(206,351)(207,350)(208,349)(209,348)(210,347)(211,346)(212,345)(213,344)(214,343)(215,342)(216,341)(217,340)(218,339)(219,338)(220,337)(221,336)(222,335)(223,334)(224,333)(225,332)(226,331)(227,330)(228,329)(229,328)(230,327)(231,326)(232,325)(233,324)(234,323)(235,322)(236,321)(237,320)(238,319)(239,318)(240,317)(241,316)(242,315)(243,314)(244,313)(245,312)(246,311)(247,310)(248,309)(249,308)(250,307)(251,306)(252,305)(253,304)(254,303)(255,302)(256,301)(257,300)(258,299)(259,298)(260,297)(261,296)(262,295)(263,294)(264,293)(265,292)(266,291)(267,290)(268,289)(269,288)(270,287)(271,286)(272,285)(273,284)(274,283)(275,282)(276,281)(277,280)(278,279)

Polycyclic presentation:

2^4,7,509|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^508

Conjugacy classes & character table

14255 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 7, 509]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^508);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);