Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C127 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C127×#1454 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C127 | C23 |
| commutator | C22 | C22×C42×C127 |
| Frattini, cosocle | C42 | C24×C127 |
| socle | C23×C127 | C2×C42 |
| prime radical | C24×C127 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C127 ◁ G | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 16256 |
A minimal faithful permutation representation — 4 generators on 161 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161);(3,4,6,10,7,9,13,16)(5,8,12,14,11,15,17,18)(19,20)(21,24)(22,26)(23,25)(27,31)(28,30)(29,32)(33,34);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18)(19,21,22,27,23,28,29,33)(20,24,26,31,25,30,32,34);(4,9)(5,11)(10,16)(12,17)(20,25)(24,30)(26,32)(31,34)14224 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 10310);