Finite Group DatabasebetaFinite Group Databasebeta| order | 24512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic6128 |
| Sylow | C383 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D6128 |
| commutator, cosocle | C16×C383 | C22 |
| Frattini | C16 | C2×D383 |
| Fitting | C32×C383 | C2 |
| socle, prime radical | C2×C383 | D16 |
| hypercenter | C32 | D383 |
| hypocenter | C383 | Q64 |
| LCS | G ▷ C16×C383 ▷ C8×C383 ▷ C4×C383 ▷ C2×C383 ▷ C383 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 447 |
| transitive degree | μt(G) = 24512 |
A minimal faithful permutation representation — 2 generators on 447 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)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(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);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,447)(67,446)(68,445)(69,444)(70,443)(71,442)(72,441)(73,440)(74,439)(75,438)(76,437)(77,436)(78,435)(79,434)(80,433)(81,432)(82,431)(83,430)(84,429)(85,428)(86,427)(87,426)(88,425)(89,424)(90,423)(91,422)(92,421)(93,420)(94,419)(95,418)(96,417)(97,416)(98,415)(99,414)(100,413)(101,412)(102,411)(103,410)(104,409)(105,408)(106,407)(107,406)(108,405)(109,404)(110,403)(111,402)(112,401)(113,400)(114,399)(115,398)(116,397)(117,396)(118,395)(119,394)(120,393)(121,392)(122,391)(123,390)(124,389)(125,388)(126,387)(127,386)(128,385)(129,384)(130,383)(131,382)(132,381)(133,380)(134,379)(135,378)(136,377)(137,376)(138,375)(139,374)(140,373)(141,372)(142,371)(143,370)(144,369)(145,368)(146,367)(147,366)(148,365)(149,364)(150,363)(151,362)(152,361)(153,360)(154,359)(155,358)(156,357)(157,356)(158,355)(159,354)(160,353)(161,352)(162,351)(163,350)(164,349)(165,348)(166,347)(167,346)(168,345)(169,344)(170,343)(171,342)(172,341)(173,340)(174,339)(175,338)(176,337)(177,336)(178,335)(179,334)(180,333)(181,332)(182,331)(183,330)(184,329)(185,328)(186,327)(187,326)(188,325)(189,324)(190,323)(191,322)(192,321)(193,320)(194,319)(195,318)(196,317)(197,316)(198,315)(199,314)(200,313)(201,312)(202,311)(203,310)(204,309)(205,308)(206,307)(207,306)(208,305)(209,304)(210,303)(211,302)(212,301)(213,300)(214,299)(215,298)(216,297)(217,296)(218,295)(219,294)(220,293)(221,292)(222,291)(223,290)(224,289)(225,288)(226,287)(227,286)(228,285)(229,284)(230,283)(231,282)(232,281)(233,280)(234,279)(235,278)(236,277)(237,276)(238,275)(239,274)(240,273)(241,272)(242,271)(243,270)(244,269)(245,268)(246,267)(247,266)(248,265)(249,264)(250,263)(251,262)(252,261)(253,260)(254,259)(255,258)(256,257)Polycyclic presentation:
2^6,383|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^382,c^b=cd,d^b=de,e^b=ef,g^b=g^3826131 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24512, 390);