Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#21781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2308 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C127×#3487 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ G | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,12,7,11,4,9)(3,13,14,22,16,8,20,24)(5,18,17,26,19,10,21,25)(15,23,30,31,29,27,28,32)(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)(2,8)(4,14)(5,15)(6,20)(7,16)(9,24)(10,23)(11,13)(12,22)(17,28)(18,27)(19,29)(21,30)(25,32)(26,31);(1,4,7,6)(2,9,11,12)(3,14,16,20)(5,17,19,21)(8,22,13,24)(10,25,18,26)(15,28,29,30)(23,31,27,32);(1,5,7,19)(2,10,11,18)(3,15,16,29)(4,17,6,21)(8,23,13,27)(9,25,12,26)(14,28,20,30)(22,31,24,32)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14769);