Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C23×C4 |
| direct product | C127×#12556 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C127 | D8 |
| commutator | C8 | C2×C42×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C23×C127 | C2×D8 |
| prime radical | C24×C127 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,22)(12,17)(13,19)(14,18)(15,21)(16,24)(20,28)(23,36)(25,30)(26,37)(27,34)(29,35)(31,39)(32,40)(33,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(2,4)(9,11,14,25,12,23,27,38,15,26,29,40,28,39,13,24)(10,16,19,31,20,32,35,37,21,33,34,36,17,30,18,22);(9,12,15,28)(10,17,21,20)(11,23,26,39)(13,14,27,29)(16,30,33,32)(18,34,35,19)(22,36,37,31)(24,25,38,40)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5544);