Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#7451 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C127 | #357 |
| commutator | C2×C16 | C2×C4×C127 |
| Frattini, cosocle | C4×C16 | C22×C127 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 2 generators on 167 points:
(2,5)(3,7)(6,8)(9,10,12,18,28,37,40,16,15,21,11,22,14,20,29,34,13,19,26,33,24,35,23,32,31,38,25,17,30,36,27,39)(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);(1,2,4,6)(3,8,7,5)(10,16)(11,23)(12,27)(14,24)(15,31)(17,20)(18,37)(19,32)(21,39)(22,36)(25,40)(26,29)(28,30)(33,35)(34,38)6223 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 439);