Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C24×C4 |
| direct product | C127×#21100 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C22×D4 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C22×C127 | C2×C4×D4 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ C23×C8×C127 ◁ G | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2)(3,4)(5,6,10,14,11,15,9,25)(7,16,20,27,18,30,19,12)(8,21,24,28,22,35,23,13)(17,31,33,36,34,26,32,29)(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)(6,12,15,27)(8,17)(9,19)(10,20)(11,18)(13,26,28,31)(14,16,25,30)(21,29,35,36)(22,34)(23,32)(24,33);(2,4)(5,8,9,23,11,22,10,24)(6,13,14,21,15,28,25,35)(7,17,19,32,18,34,20,33)(12,26,16,29,27,31,30,36);(7,18)(8,22)(12,27)(13,28)(16,30)(19,20)(21,35)(23,24)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14088);