Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#18440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C127×#3559 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
904 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)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,23,20)(19,27,30,32)(21,26,22,25)(24,31,28,29)(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)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,27)(26,32);(1,3,5,10)(2,7,9,14)(4,11,12,16)(6,13,8,15)(17,20,23,18)(19,28,30,24)(21,25,22,26)(27,31,32,29);(3,10)(7,14)(11,16)(13,15)(17,21,23,22)(18,25,20,26)(19,29,30,31)(24,27,28,32)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11428);