Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#10783 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #386 |
| commutator | C2×C4 | C2×C42×C127 |
| Frattini, cosocle | C23×C4 | C23×C127 |
| socle | C2×C127 | C22⋊#95 |
| prime radical | C22×#41×C127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C23×C4×C127 ◁ G | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2)(3,7,12,21)(4,14,16,24)(5,10)(6,9)(8,15,22,17)(11,25)(13,20,28,32)(18,23)(19,29)(26,31)(27,30)(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)(2,7,9,21)(4,11)(5,13)(6,12)(8,19,22,30)(10,20,23,32)(14,25,24,31)(15,27)(16,26)(17,29)(18,28);(1,4)(2,8)(3,11)(5,17)(6,16)(7,19)(9,22)(10,24)(12,26)(13,29)(14,23)(15,18)(20,31)(21,30)(25,32)(27,28)7366 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 3771);