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×#17610 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C127×#3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×#117×C127 ◁ G | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2)(3,11,14,7)(4,8)(5,9)(6,10)(12,16,13,15)(17,18)(19,28)(20,25)(21,26)(22,31)(23,27)(24,29)(30,32)(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);(1,3,6,14)(2,7,10,11)(4,12,5,13)(8,15,9,16)(17,19,23,20)(18,24,27,32)(21,29,22,30)(25,31,28,26);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(17,20,23,19)(18,25,27,28)(21,29,22,30)(24,31,32,26);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)(17,21,23,22)(18,26,27,31)(19,29,20,30)(24,25,32,28)6604 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 10598);