Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#12554 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C127 | C2×D4 |
| commutator | C4 | C2×C4×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C2×C22⋊C8 |
| prime radical | C24×C127 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C127 ◁ C23×C8×C127 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(5,6,9,15,10,16,24,33,11,13,25,29,8,14,22,31)(7,17,19,34,18,27,35,32,20,30,23,36,21,12,26,28)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,2,3,4)(5,7)(6,12)(8,21)(9,23)(10,18)(11,20)(13,27)(14,30)(15,32)(16,17)(19,25)(22,35)(24,26)(28,29)(31,34)(33,36);(6,13)(12,27)(14,16)(15,29)(17,30)(28,32)(31,33)(34,36)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5542);