Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#21779 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2308 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C127×#3485 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ G | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,5,10,6,9,18,24)(3,11,14,25,13,7,17,20)(4,8,15,23,16,22,29,32)(12,19,27,30,26,21,28,31)(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);(1,3)(2,7)(4,12)(5,17)(6,13)(8,21)(9,11)(10,25)(14,18)(15,28)(16,26)(19,22)(20,24)(23,30)(27,29)(31,32);(1,4,6,16)(2,8,9,22)(3,12,13,26)(5,15,18,29)(7,19,11,21)(10,23,24,32)(14,27,17,28)(20,30,25,31);(2,9)(5,18)(7,11)(8,22)(14,17)(15,29)(19,21)(27,28)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14767);