Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#9371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #386 |
| commutator | C42 | C22×C4×C127 |
| Frattini, cosocle | C2×C42 | C23×C127 |
| prime radical | C127×#3214 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18,22,24,21,20,26,29)(19,25,27,32,28,23,30,31)(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)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19,26,27,21,28,22,30)(18,23,24,32,20,25,29,31);(2,8)(4,5)(7,15)(11,13)(17,20)(18,21)(19,25)(22,24)(23,28)(26,29)(27,31)(30,32)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 2359);