Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#7177 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | #102 |
| commutator | C8 | C4×C8×C127 |
| Frattini, cosocle | C2×C4×C8 | C22×C127 |
| socle | C22×C127 | #340 |
| prime radical | C23×C127 | #103 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G | |
238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 2 generators on 167 points:
(1,2,4,7,5,8,3,6)(9,10,13,18,15,21,32,20)(11,22,24,35,27,37,26,40)(12,28,30,17,31,19,14,33)(16,25,34,23,36,39,38,29)(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);(2,6,8,7)(9,11,14,25,32,24,31,23,15,27,30,39,13,26,12,29)(10,16,19,35,18,38,33,22,21,36,28,40,20,34,17,37)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 165);