Finite Group DatabasebetaFinite Group Databasebeta| order | 30656 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D15328 |
| Fitting | C32×C479 ⋊ C2 |
| Sylow | C479 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7664 |
| commutator, cosocle | C16×C479 | C22 |
| Frattini | C16 | C2×D479 |
| Fitting | C32×C479 | C2 |
| socle | C2×C479 | D16 |
| hypercenter | C32 | D479 |
| hypocenter | C479 | D32 |
| LCS | G ▷ C16×C479 ▷ C8×C479 ▷ C4×C479 ▷ C2×C479 ▷ C479 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 511 |
| transitive degree | μt(G) = 15328 |
A minimal faithful permutation representation — 2 generators on 511 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,511)(35,510)(36,509)(37,508)(38,507)(39,506)(40,505)(41,504)(42,503)(43,502)(44,501)(45,500)(46,499)(47,498)(48,497)(49,496)(50,495)(51,494)(52,493)(53,492)(54,491)(55,490)(56,489)(57,488)(58,487)(59,486)(60,485)(61,484)(62,483)(63,482)(64,481)(65,480)(66,479)(67,478)(68,477)(69,476)(70,475)(71,474)(72,473)(73,472)(74,471)(75,470)(76,469)(77,468)(78,467)(79,466)(80,465)(81,464)(82,463)(83,462)(84,461)(85,460)(86,459)(87,458)(88,457)(89,456)(90,455)(91,454)(92,453)(93,452)(94,451)(95,450)(96,449)(97,448)(98,447)(99,446)(100,445)(101,444)(102,443)(103,442)(104,441)(105,440)(106,439)(107,438)(108,437)(109,436)(110,435)(111,434)(112,433)(113,432)(114,431)(115,430)(116,429)(117,428)(118,427)(119,426)(120,425)(121,424)(122,423)(123,422)(124,421)(125,420)(126,419)(127,418)(128,417)(129,416)(130,415)(131,414)(132,413)(133,412)(134,411)(135,410)(136,409)(137,408)(138,407)(139,406)(140,405)(141,404)(142,403)(143,402)(144,401)(145,400)(146,399)(147,398)(148,397)(149,396)(150,395)(151,394)(152,393)(153,392)(154,391)(155,390)(156,389)(157,388)(158,387)(159,386)(160,385)(161,384)(162,383)(163,382)(164,381)(165,380)(166,379)(167,378)(168,377)(169,376)(170,375)(171,374)(172,373)(173,372)(174,371)(175,370)(176,369)(177,368)(178,367)(179,366)(180,365)(181,364)(182,363)(183,362)(184,361)(185,360)(186,359)(187,358)(188,357)(189,356)(190,355)(191,354)(192,353)(193,352)(194,351)(195,350)(196,349)(197,348)(198,347)(199,346)(200,345)(201,344)(202,343)(203,342)(204,341)(205,340)(206,339)(207,338)(208,337)(209,336)(210,335)(211,334)(212,333)(213,332)(214,331)(215,330)(216,329)(217,328)(218,327)(219,326)(220,325)(221,324)(222,323)(223,322)(224,321)(225,320)(226,319)(227,318)(228,317)(229,316)(230,315)(231,314)(232,313)(233,312)(234,311)(235,310)(236,309)(237,308)(238,307)(239,306)(240,305)(241,304)(242,303)(243,302)(244,301)(245,300)(246,299)(247,298)(248,297)(249,296)(250,295)(251,294)(252,293)(253,292)(254,291)(255,290)(256,289)(257,288)(258,287)(259,286)(260,285)(261,284)(262,283)(263,282)(264,281)(265,280)(266,279)(267,278)(268,277)(269,276)(270,275)(271,274)(272,273)Polycyclic presentation:
2^6,479|c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^478,c^b=cd,d^b=de,e^b=ef,g^b=g^4787667 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30656, 385);