Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C127 · Schur multiplier (H2) = C24 |
| direct product | C127×#19228 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C127 |
| prime radical | #121×C127 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,9)(3,10,13,16)(4,8,5,14)(7,11,15,12)(17,18)(19,26)(20,25)(21,23)(22,29)(24,27)(28,32)(30,31)(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);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(18,23)(24,31)(25,29)(26,32);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,19)(18,24)(20,27)(21,28)(22,30)(23,31)(25,32)(26,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,20,21,22)(18,25,23,29)(19,27,28,30)(24,32,31,26)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 12216);