Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13558 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C127 | C2×D4 |
| commutator | C4 | C22×C16×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C2×C127 | C2×C22⋊C16 |
| prime radical | C2×#41×C127 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C127 ◁ C22×C16×C127 ◁ G | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,5,12,6,13,24,39,7,14,25,40,27,41,52,61,8,10,26,33,28,34,53,57,4,11,21,36,22,37,49,59)(3,15,17,42,18,43,46,62,16,30,44,38,45,55,63,60,19,35,23,50,47,58,51,64,20,9,29,31,48,32,54,56)(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)(2,9)(4,20)(5,23)(6,18)(7,16)(8,19)(10,30)(11,35)(12,38)(13,32)(14,15)(17,26)(21,44)(22,48)(24,51)(25,29)(27,45)(28,47)(31,33)(34,55)(36,42)(37,58)(39,60)(40,50)(41,43)(46,53)(49,63)(52,54)(56,57)(59,62)(61,64);(2,10)(9,30)(11,14)(12,33)(13,34)(15,35)(31,38)(32,55)(36,40)(37,41)(39,57)(42,50)(43,58)(56,60)(59,61)(62,64)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6546);