Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13551 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C127 | C42 |
| commutator | C4 | C2×C4×C8×C127 |
| Frattini, cosocle | C2×C16 | C23×C127 |
| socle | C2×C127 | C2×#346 |
| prime radical | C23×C127 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C16×C127 ◁ G | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 191 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 191 points:
(1,2,4,9,5,11,19,31,6,12,20,32,22,35,46,57,7,10,21,30,23,33,47,55,24,34,48,56,49,58,63,64)(3,13,15,37,16,38,41,60,17,26,42,29,43,52,61,54,14,36,18,44,39,59,45,62,40,8,25,27,50,28,51,53)(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)(2,8,10,26)(4,18)(5,16)(6,17)(7,14)(9,29,30,27)(11,28,33,52)(12,13,34,36)(15,21)(19,45)(20,25)(22,43)(23,39)(24,40)(31,54,55,53)(32,44,56,37)(35,38,58,59)(41,47)(42,48)(46,51)(49,50)(57,62,64,60)(61,63);(3,14)(8,26)(13,36)(15,18)(16,39)(17,40)(25,42)(27,29)(28,52)(37,44)(38,59)(41,45)(43,50)(51,61)(53,54)(60,62)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6539);