Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#11939 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #102 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C4×C8 | C23×C127 |
| socle | C22×C127 | C2×#102 |
| prime radical | C23×C127 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 3 generators on 167 points:
(2,5)(3,7)(6,8)(9,10,14,19,12,17,27,18,15,20,30,35,28,36,13,29)(11,21,24,33,26,37,40,31,25,38,39,16,22,32,23,34)(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);(1,2,4,6)(3,8,7,5)(9,11,15,25)(10,16,20,33)(12,22,28,26)(13,24,27,39)(14,23,30,40)(17,31,36,34)(18,37,29,32)(19,38,35,21);(9,12,15,28)(10,17,20,36)(11,22,25,26)(13,14,27,30)(16,31,33,34)(18,35,29,19)(21,32,38,37)(23,39,40,24)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 4927);