Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C127 · Schur multiplier (H2) = C23 |
| direct product | C127×#13442 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #2054 |
| commutator, Frattini, cosocle | C22×C8 | C23×C127 |
| prime radical | C2×#509×C127 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 3 generators on 159 points:
(2,9)(3,15)(4,20)(5,6)(7,21)(11,24)(12,14)(13,22)(16,25)(17,31)(18,23)(19,29)(26,30)(28,32)(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);(1,2)(3,16)(4,11)(5,12)(6,22)(7,24)(8,14)(9,20)(10,17)(13,21)(15,26)(18,32)(19,28)(23,27)(25,31)(29,30);(1,3,4,17,21,31,5,15,8,19,6,23,7,18,20,29)(2,10,9,16,24,30,22,32,14,27,12,28,13,26,11,25)4318 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 6430);