Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C24×C4 |
| direct product | C127×#18306 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C127 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(2,7)(3,11)(4,15)(8,20)(9,18)(10,16)(13,27)(14,25)(17,26)(19,22)(21,30)(23,32)(24,29)(28,31)(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)(3,8)(4,16)(5,10)(6,9)(7,15)(11,22)(12,21)(13,28)(14,23)(17,30)(18,26)(19,29)(20,27)(24,32)(25,31);(4,17)(7,18)(13,24)(15,26)(16,30)(19,20)(27,29)(28,32);(1,3,12,25)(2,8,21,31)(4,13,26,27)(5,14,6,11)(7,19,30,32)(9,22,10,23)(15,29,17,24)(16,28,18,20)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11294);