Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C127 · Schur multiplier (H2) = C22×C4 |
| direct product | C127×#12651 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | #442 |
| commutator | C8 | C2×C42×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| socle | C2×C127 | #1762 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C2×#97×C127 ◁ G | |
412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,5,12,8,13,7,23)(3,14,17,29,18,30,21,25)(4,10,6,22,20,28,19,11)(9,15,24,16,27,31,26,32)(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);(1,3,6,15,5,21,20,32,8,18,19,31,7,17,4,16)(2,9,10,25,23,24,11,14,13,27,28,29,12,26,22,30);(1,4,5,6,8,20,7,19)(2,10,12,22,13,28,23,11)(3,15,17,16,18,31,21,32)(9,14,24,29,27,30,26,25)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5639);