Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13161 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C127 | C2×D4 |
| commutator | C2×C4 | C22×C8×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C8×D4 |
| prime radical | C23×C127 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C127 ◁ C22×C16×C127 ◁ G | |
298 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)(5,6,10,16,11,15,25,31,12,17,24,14,9,26,8,23)(7,13,20,29,21,28,34,36,22,30,33,27,19,35,18,32)(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)(5,7,10,20,11,21,25,34,12,22,24,33,9,19,8,18)(6,13,16,29,15,28,31,36,17,30,14,27,26,35,23,32);(5,8,9,24,12,25,11,10)(6,14,15,23,17,16,26,31)(7,18,19,33,22,34,21,20)(13,27,28,32,30,29,35,36)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6149);