Finite Group DatabasebetaFinite Group Databasebeta| order | 27008 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C211 · Schur multiplier (H2) = C2 |
| direct product | #1401×C211 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C211 | D4 |
| commutator | C4 | C2×C16×C211 |
| Frattini, cosocle | C2×C16 | C22×C211 |
| socle | C2×C211 | C22⋊C16 |
| prime radical | #41×C211 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C211 ◁ C2×C16×C211 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 275 |
| transitive degree | μt(G) = 13504 |
A minimal faithful permutation representation — 2 generators on 275 points:
(1,2,5,11,6,12,24,37,7,13,25,38,27,40,52,60,8,14,26,39,28,41,53,61,4,10,21,34,22,35,49,58)(3,15,17,42,18,43,46,62,16,32,44,36,45,56,63,59,19,33,23,50,47,57,51,64,20,9,29,30,48,31,54,55)(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);(1,3)(2,9)(4,20)(5,23)(6,18)(7,16)(8,19)(10,33)(11,36)(12,31)(13,15)(14,32)(17,26)(21,44)(22,48)(24,51)(25,29)(27,45)(28,47)(30,39)(34,42)(35,57)(37,59)(38,50)(40,43)(41,56)(46,53)(49,63)(52,54)(55,61)(58,62)(60,64)11816 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27008, 133);