Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C2×C4 |
| direct product | C127×#13156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1936 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#135×C127 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(2,8,12,19)(3,14,17,22)(4,9)(5,20)(6,23)(10,25)(11,28,21,13)(16,30,26,18)(24,29)(27,31)(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);(1,2,9,11,23,28,4,19,7,12,5,21,6,13,20,8)(3,15,22,31,30,10,26,27,17,32,14,24,18,25,16,29);(1,3,4,16,6,18,9,14,7,17,20,26,23,30,5,22)(2,10,8,24,13,15,21,27,12,25,19,31,28,32,11,29)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6144);