Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#12934 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1853 |
| commutator | C2×C8 | C22×C4×C127 |
| Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#41×C127 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(1,2,5,12,7,13,6,11)(3,14,17,26,18,29,16,9)(4,19,21,28,23,27,22,10)(8,20,24,15,25,31,30,32)(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)(2,8,13,25)(4,20)(5,16)(6,17)(7,18)(9,10,26,28)(11,24,12,30)(14,27,29,19)(15,22)(21,32)(23,31);(1,3)(2,9)(4,15)(5,17)(6,16)(7,18)(8,10)(11,29)(12,14)(13,26)(19,24)(20,22)(21,31)(23,32)(25,28)(27,30)5080 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 5922);