Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C25 |
| direct product | C127×#18240 |
| 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 | |
1056 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)(4,14)(8,19)(9,18)(10,15)(12,23)(16,28)(20,25)(21,30)(22,29)(26,27)(31,32)(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)(3,8)(4,15)(5,10)(6,9)(7,14)(11,20)(12,25)(13,21)(16,29)(17,22)(18,28)(19,23)(24,32)(26,30)(27,31);(3,11)(4,16)(7,18)(8,20)(12,26)(13,24)(14,28)(15,29)(19,31)(21,32)(23,27)(25,30);(1,3,6,13)(2,8,9,21)(4,12,16,27)(5,11,17,24)(7,19,18,30)(10,20,22,32)(14,23,28,26)(15,25,29,31)4699 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11228);