Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C27 |
| direct product | C127×#21017 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #386 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C2×#386 |
| prime radical | C127×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,12,7,11,5,10)(3,8,14,22,15,23,19,25)(4,16,17,26,18,9,20,24)(13,21,29,32,28,30,27,31)(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);(1,3,6,19,7,15,5,14)(2,8,10,22,11,23,12,25)(4,13,20,29,18,28,17,27)(9,21,24,31,16,30,26,32);(1,4)(2,9)(3,13)(5,17)(6,20)(7,18)(8,21)(10,24)(11,16)(12,26)(14,27)(15,28)(19,29)(22,31)(23,30)(25,32);(1,5,7,6)(2,10,11,12)(3,14,15,19)(4,17,18,20)(8,22,23,25)(9,24,16,26)(13,27,28,29)(21,31,30,32)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14005);