Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C24×C4 |
| direct product | C127×#18334 |
| 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 | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(2,8)(3,13)(4,17)(5,14)(9,23)(10,27)(11,20)(12,25)(15,19)(16,18)(21,30)(24,26)(28,31)(29,32)(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);(1,2)(3,9)(4,10)(5,20)(6,12)(7,11)(8,14)(13,24)(15,28)(16,27)(17,26)(18,31)(19,23)(21,32)(22,29)(25,30);(1,3,7,16)(2,9,11,27)(4,14,19,5)(6,13,22,18)(8,23,20,10)(12,24,29,31)(15,30,17,21)(25,26,32,28);(1,4,6,15)(2,10,12,28)(3,14,13,30)(5,18,21,16)(7,19,22,17)(8,24,25,9)(11,23,29,26)(20,31,32,27)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11322);