Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C127 · Schur multiplier (H2) = C22×C42 |
| direct product | C127×#10699 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C127 | C42 |
| commutator | C4 | C43×C127 |
| Frattini, cosocle | C23×C4 | C23×C127 |
| socle | C22×C127 | #376 |
| prime radical | C24×C127 | #41 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C127 ◁ G | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2)(3,4)(5,6)(7,17,20,27)(8,13)(9,14)(10,16)(11,15)(12,19,26,21)(18,31,33,36)(22,28)(23,30)(24,29)(25,32,35,34)(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);(2,4)(5,7)(6,12,15,26)(8,18)(9,19)(10,21)(11,20)(13,25,29,35)(14,17,16,27)(22,32)(23,34)(24,33)(28,31,30,36);(5,8,10,23,11,24,9,22)(6,13,16,30,15,29,14,28)(7,18,21,34,20,33,19,32)(12,25,27,36,26,35,17,31)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 3687);