Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#18298 |
| 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 ◁ C23×C4×C127 ◁ G | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(2,9)(3,14)(4,19)(5,15)(10,28)(11,30)(12,21)(13,27)(16,20)(17,18)(22,23)(24,32)(25,26)(29,31)(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)(3,10)(4,11)(5,21)(6,24)(7,12)(8,13)(9,15)(14,30)(16,31)(17,26)(18,29)(19,25)(20,28)(22,27)(23,32);(1,3,8,18)(2,10,13,29)(4,15,19,22)(5,20,23,16)(6,14,7,17)(9,25,27,11)(12,26,24,30)(21,28,32,31);(1,4,7,16)(2,11,12,31)(3,15,17,5)(6,20,8,19)(9,26,21,10)(13,25,24,28)(14,23,18,22)(27,30,32,29)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11286);