Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#21677 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C127 | C24 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle, prime radical | C23×C127 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C127 ◁ G | |
618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,20,22,24,29,28)(19,23,27,26,25,30,32,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(19,25)(23,30)(26,31)(27,32);(1,4,5,3)(2,8,6,7)(17,20,21,24,22,28,29,18)(19,26,27,30,25,31,32,23)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14665);