Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#13470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C22×D8×C127 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 163 points:
(5,6,11,28,12,18,26,17)(7,19,24,31,25,35,36,33)(8,14,20,15,9,16,10,27)(13,29,34,21,30,23,32,22)(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);(1,2)(3,4)(5,7,20,29,11,24,9,21,12,25,10,23,26,36,8,22)(6,13,14,33,17,32,27,35,18,30,16,31,28,34,15,19);(2,4)(6,14)(7,21)(8,20)(9,10)(11,26)(13,30)(15,17)(16,18)(19,31)(22,25)(23,36)(24,29)(27,28)(33,35)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6458);