Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C127 · Schur multiplier (H2) = C2×C4 |
| direct product | C127×#13124 |
| 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 | C2×#41×C127 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C127 ◁ C22×C16×C127 ◁ G | |
332 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)(5,6,11,17,10,15,8,26,12,18,27,14,9,16,20,33)(7,19,24,35,23,32,21,29,25,30,36,34,22,13,28,31)(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);(5,7,20,28,9,22,27,36,12,25,8,21,10,23,11,24)(6,13,26,35,16,30,17,31,18,32,33,34,15,19,14,29);(2,4)(5,8,9,11,12,20,10,27)(6,14,15,33,18,17,16,26)(7,21,22,24,25,28,23,36)(13,29,19,34,32,31,30,35)11176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6112);