Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#10467 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23×C127 |
| prime radical | #532×C127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,25)(13,30)(14,18)(15,19)(16,22)(17,28)(21,36)(23,32)(24,35)(26,31)(27,34)(29,39)(33,38)(37,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(2,5)(6,7)(9,11,15,24)(10,16,19,32)(12,26,28,37)(13,23,14,22)(17,33,25,36)(18,35,30,20)(21,27,38,29)(31,39,40,34);(2,6)(4,8)(5,7)(9,12,13,27,15,28,14,29)(10,17,18,34,19,25,30,39)(11,21,22,37,24,38,23,26)(16,31,20,33,32,40,35,36)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 3455);