Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C127 · Schur multiplier (H2) = C23×C4 |
| direct product | C127×#17341 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C127 | C24 |
| commutator | C22 | C22×C42×C127 |
| Frattini, cosocle | C42 | C24×C127 |
| socle | C22×C127 | C22×C42 |
| prime radical | C2×#41×C127 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C127 ◁ G | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32)(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);(3,9)(6,12)(10,15)(13,16)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(19,25)(22,28)(26,31)(29,32);(3,9)(6,12)(10,15)(13,16)(18,23)(22,28)(24,30)(29,32)12700 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 10329);