Finite Group DatabasebetaFinite Group Databasebeta| order | 27008 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C211 · Schur multiplier (H2) = C2×C4 |
| direct product | #1384×C211 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C211 | #106 |
| commutator | C8 | C42×C211 |
| Frattini, cosocle | C22×C8 | C22×C211 |
| socle | C2×C211 | #340 |
| prime radical | C23×C211 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C211 ◁ C22×C4×C211 ◁ C22×C8×C211 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 243 |
| transitive degree | μt(G) = 6752 |
A minimal faithful permutation representation — 2 generators on 243 points:
(1,2)(3,13,15,27)(4,10)(5,20)(6,22)(7,12)(8,14,23,28)(9,18)(11,21)(16,31,29,25)(17,32,30,24)(19,26)(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);(1,3,5,16,21,30,19,28,7,15,18,29,6,17,4,14)(2,8,10,24,22,31,9,13,12,23,26,32,11,25,20,27)6752 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27008, 116);