Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C22 | C2×D32 |
| prime radical | C2×D32 | C22 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
121 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 121 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2,3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68);(6,36)(7,67)(8,34)(9,65)(10,32)(11,63)(12,30)(13,61)(14,28)(15,59)(16,26)(17,57)(18,24)(19,55)(20,22)(21,53)(23,51)(25,49)(27,47)(29,45)(31,43)(33,41)(35,39)(38,68)(40,66)(42,64)(44,62)(46,60)(48,58)(50,56)(52,54)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60786);