Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1804 |
| 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 | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,4,10,7,12,6,24)(3,13,16,8,17,27,14,28)(5,19,18,25,23,32,22,11)(9,21,30,20,26,31,29,15)(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)(2,8,12,28)(4,14)(5,20)(6,16)(7,17)(9,25,26,11)(10,13,24,27)(15,23)(18,21)(19,29,32,30)(22,31);(1,3)(2,9)(4,16)(5,21)(6,14)(7,17)(8,25)(10,30)(11,28)(12,26)(13,19)(15,22)(18,20)(23,31)(24,29)(27,32)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5816);