Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#17598 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C127×#3520 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C4×D4×C127 ◁ G | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(2,7,9,21)(3,12)(4,15)(8,23,25,31)(10,22,11,16)(13,27)(14,30)(17,19)(20,24,32,26)(28,29)(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);(1,2,5,10)(3,8,13,24)(4,16,15,21)(6,11,18,9)(7,19,22,17)(12,26,27,25)(14,23,29,32)(20,30,31,28);(1,3)(2,8)(4,14)(5,13)(6,12)(7,20)(9,25)(10,24)(11,26)(15,29)(16,23)(17,28)(18,27)(19,30)(21,32)(22,31);(3,13)(8,24)(12,27)(14,29)(20,31)(23,32)(25,26)(28,30)6223 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 10586);