Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C127 · Schur multiplier (H2) = C1 |
| direct product | C127×#7454 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1365 |
| commutator | C2×C16 | C2×C4×C127 |
| Frattini, cosocle | C4×C16 | C22×C127 |
| prime radical | C22×C127 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 2 generators on 191 points:
(1,2,5,12,30,14,25,11,6,33,31,43,36,42,28,62,9,15,26,47,7,34,27,44,8,13,32,46,24,49,4,23)(3,16,18,38,57,41,56,50,19,45,58,10,35,37,17,39,22,53,60,48,20,61,55,51,21,52,59,40,54,64,29,63)(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);(1,3)(2,10,15,40)(4,18)(5,29)(6,21)(7,35)(8,19)(9,22)(11,41,44,61)(12,45,47,52)(13,48,33,38)(14,50,34,51)(16,46,53,43)(17,26)(20,36)(23,37,62,64)(24,57)(25,59)(27,58)(28,60)(30,54)(31,55)(32,56)(39,42,63,49)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 442);