Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#10974 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #386 |
| commutator | C24 | C22×C4×C127 |
| Frattini, cosocle | C23×C4 | C23×C127 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G | |
1258 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)(5,6,9,14,10,13,22,28)(7,15,17,29,18,11,21,24)(8,12,20,27,19,26,32,36)(16,25,31,35,30,23,33,34)(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);(5,7,10,18)(6,11,13,15)(8,16,19,30)(9,21,22,17)(12,25,26,23)(14,29,28,24)(20,33,32,31)(27,34,36,35);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,26)(14,27)(15,25)(17,31)(18,30)(21,33)(22,32)(24,34)(28,36)(29,35)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 3962);