Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C22×C4 |
| direct product | C127×#12633 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | #99 |
| commutator | C2×C4 | C2×C42×C127 |
| Frattini, cosocle | C2×#41 | C23×C127 |
| socle | C2×C127 | C4×#99 |
| prime radical | #532×C127 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C2×C8×C127 ◁ C2×#131×C127 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,6,11,7,4,9,18)(3,12,15,27,16,28,22,32)(5,17,19,31,20,10,24,25)(8,21,23,30,13,26,29,14)(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)(4,13)(5,14,20,30)(6,15)(7,16)(9,22)(10,12,17,28)(11,23)(18,29)(19,21,24,26)(25,27,31,32);(1,4,6,18,7,2,9,11)(3,13,15,29,16,8,22,23)(5,17,19,31,20,10,24,25)(12,26,27,14,28,21,32,30)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5621);