Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#8512 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #379 |
| commutator | C42 | C22×C4×C127 |
| Frattini, cosocle | C2×C42 | C23×C127 |
| prime radical | #121×C127 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 159 points:
(2,6,8,11)(7,15,16,14)(18,23)(19,26)(20,21)(25,28)(27,29)(30,31)(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);(1,2)(3,7)(4,11)(5,8)(6,13)(9,14)(10,16)(12,15)(17,18)(19,24)(20,28)(21,23)(22,25)(26,31)(27,32)(29,30);(1,3,5,10)(2,7,8,16)(4,12,13,9)(6,14,11,15)(17,19,22,27)(18,24,25,32)(20,26,21,29)(23,30,28,31)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 1500);