Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C127 · Schur multiplier (H2) = C26 |
| direct product | C127×#21785 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2308 |
| commutator | C2×C4 | C23×C4×C127 |
| Frattini, cosocle | C22×C4 | C24×C127 |
| prime radical | C127×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ G | |
1262 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)(3,13,17,8)(4,10,5,19)(6,12)(7,11)(9,18,27,20)(14,30)(15,24)(16,29,21,25)(22,28)(23,31)(26,32)(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)(4,14)(5,15)(6,21)(7,17)(9,26)(10,24)(11,13)(12,29)(16,22)(18,32)(19,30)(20,31)(23,27)(25,28);(1,4,7,5)(2,9,11,27)(3,14,17,15)(6,20,22,18)(8,23,13,26)(10,12,19,28)(16,32,21,31)(24,25,30,29);(1,5,7,4)(2,10,11,19)(3,15,17,14)(6,20,22,18)(8,24,13,30)(9,12,27,28)(16,32,21,31)(23,25,26,29)5842 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 14773);