Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C24×C4 |
| direct product | C127×#18461 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | #447×C127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2)(3,9)(4,6)(5,13)(7,11)(8,15)(10,14)(12,16)(20,27)(22,28)(23,30)(25,31)(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);(1,3,6,12)(2,7,4,10)(5,14,15,11)(8,16,13,9)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(1,4,6,2)(3,10,12,7)(5,13,15,8)(9,11,16,14)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(2,4)(7,10)(8,13)(9,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 11449);