Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C22 |
| direct product | C127×#12657 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C127 | C2×C42 |
| commutator | C2×C4 | C2×C42×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C22×C127 | C2×#97 |
| prime radical | C23×C127 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C127 ◁ G | |
334 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)(7,17,20,32)(8,13)(9,14)(10,25)(11,16)(12,19,28,33)(15,26)(18,29,21,35)(22,36,34,27)(23,30)(24,31)(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);(2,4)(5,7,10,21,9,19,23,34,11,20,26,18,24,33,8,22)(6,12,15,27,14,17,13,29,16,28,25,36,31,32,30,35);(5,8,9,10,11,23,24,26)(6,13,14,25,16,30,31,15)(7,18,33,34,20,21,19,22)(12,27,32,35,28,36,17,29)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5645);