Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7491 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
700 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 700 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(7,8,11,15,12,9,10,18)(13,16,20,17,22,19,21,14);(3,4,5,6)(7,9,12,8)(10,14,11,17)(13,16,22,19)(15,21,18,20)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59003);