Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#18262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C127×#3534 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,4)(3,11,15,27)(5,18,20,30)(7,9,22,16)(8,23,12,13)(10,25,17,21)(14,26,29,31)(19,24,32,28)(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)(2,8)(4,12)(5,19)(6,15)(7,14)(9,24)(10,26)(11,18)(13,21)(16,28)(17,31)(20,32)(22,29)(23,25)(27,30);(1,4)(2,6)(3,12)(5,10)(7,16)(8,15)(9,22)(11,13)(14,28)(17,20)(18,21)(19,26)(23,27)(24,29)(25,30)(31,32);(2,9)(4,16)(8,24)(10,18)(11,26)(12,28)(17,30)(27,31)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11250);