Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C127 · Schur multiplier (H2) = C1 |
| direct product | C127×#7093 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C127 | #331 |
| commutator | C4×C8 | C2×C4×C127 |
| Frattini, cosocle | C82 | C22×C127 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C42×C127 ◁ C82×C127 ◁ G | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 32512 |
A minimal faithful permutation representation — 2 generators on 159 points:
(2,7,8,15)(3,11)(4,5)(9,12,10,16)(13,14)(17,18,20,24,30,29,32,27,21,25,31,23,22,19,26,28)(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);(1,2,6,8)(3,12,14,16)(4,10,5,9)(7,11,15,13)(17,19)(18,22)(20,24)(21,29)(23,31)(25,30)(26,27)(28,32)5461 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 81);