Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#9304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | #386 |
| commutator | C42 | C22×C4×C127 |
| Frattini, cosocle | C2×C42 | C23×C127 |
| prime radical | C23×C127 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8128 |
A minimal faithful permutation representation — 3 generators on 167 points:
(2,6)(3,4)(7,8)(9,10)(11,15,20,30)(12,22)(13,25)(14,17)(16,23)(18,27)(19,33,34,40)(21,37,36,29)(24,32)(26,31)(28,38,39,35)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(1,2,4,6,5,8,3,7)(9,11)(10,15,17,30)(12,19)(13,21)(14,20)(16,28,31,39)(18,29,25,37)(22,33,32,40)(23,35)(24,34)(26,38)(27,36);(1,3,5,4)(2,7,8,6)(9,12,14,24)(10,16,17,31)(11,19,20,34)(13,26,27,23)(15,28,30,39)(18,32,25,22)(21,38,36,35)(29,40,37,33)6223 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 2292);