Finite Group DatabasebetaFinite Group Databasebeta| order | 32512 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C127 · Schur multiplier (H2) = C2 |
| direct product | C127×#7526 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C127 | #1404 |
| commutator | #125 | C2×C4×C127 |
| Frattini, cosocle | #491 | C22×C127 |
| prime radical | C22×C127 | AΣL(1,𝔽16) |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C127 ◁ C22×C127 ◁ C2×C4×C127 ◁ C42×C127 ◁ #491×C127 ◁ G | |
126 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 159 |
| transitive degree | μt(G) = 4064 |
A minimal faithful permutation representation — 2 generators on 159 points:
(1,2,5,10,6,12,3,13)(4,16,18,26,20,27,32,8)(7,11,15,30,22,31,21,29)(9,17,25,23,28,14,24,19)(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);(1,3,6,5)(2,8,12,26)(4,17,20,14)(7,21,22,15)(9,13,28,10)(11,25,31,24)(16,29,27,30)(18,19,32,23)2413 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32512, 514);