Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#11872 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1519 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C2×M16 | C23×C127 |
| prime radical | C127×#3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×#131×C127 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,5,11,7,13,6,12)(3,14,17,9,18,27,15,28)(4,8,21,24,22,25,19,10)(16,29,31,23,20,26,32,30)(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);(2,8)(3,15)(4,19)(5,6)(9,26)(10,12)(11,24)(13,25)(14,23)(17,18)(21,22)(27,30)(28,29)(31,32);(1,3,6,15,7,18,5,17)(2,9,11,27,13,28,12,14)(4,20,21,32,22,16,19,31)(8,23,10,29,25,30,24,26)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4860);