Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#21483 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C127 | #120 |
| commutator | C23 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| socle | C2×C127 | C4×#120 |
| prime radical | C127×#3025 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 4 generators on 159 points:
(1,2,6,11,7,5,10,19)(3,8,15,22,14,13,21,28)(4,9,18,24,17,16,23,29)(12,25,27,32,26,20,30,31)(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)(2,8)(4,12)(5,13)(6,15)(7,14)(9,20)(10,21)(11,22)(16,25)(17,26)(18,27)(19,28)(23,30)(24,31)(29,32);(1,4)(2,9)(3,12)(5,16)(6,18)(7,17)(8,20)(10,23)(11,24)(13,25)(14,26)(15,27)(19,29)(21,30)(22,31)(28,32);(1,5,6,19,7,2,10,11)(3,13,15,28,14,8,21,22)(4,16,18,29,17,9,23,24)(12,25,27,32,26,20,30,31)8128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14471);