Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#18331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C127×#3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,7,13)(3,15,17,28)(4,11,5,21)(6,12,8,14)(9,22,26,16)(10,20,23,19)(18,25,24,32)(27,31,29,30)(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);(1,3)(2,9)(4,16)(5,22)(6,24)(7,17)(8,18)(10,25)(11,28)(12,29)(13,26)(14,27)(15,21)(19,31)(20,30)(23,32);(1,2,8,14)(3,9,18,27)(4,10,20,21)(5,23,19,11)(6,12,7,13)(15,16,25,30)(17,26,24,29)(22,32,31,28);(1,4,7,5)(2,10,13,23)(3,16,17,22)(6,19,8,20)(9,25,26,32)(11,14,21,12)(15,29,28,27)(18,30,24,31)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11319);