Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#11945 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#102 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| prime radical | #466×C127 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,21,25,30,23,32,28,22,26,31,24,20,19,27,29)(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);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,20)(21,25)(22,23)(24,31)(26,30)(27,28)(29,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,30)(18,23,26,19)(21,27,31,32)(24,29,25,28)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4933);