Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | #407 | C23 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
1564 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,6,11,8,9,15,16,7,12,14,18,5,10,13,17);(4,9)(5,8)(6,14)(10,12)(11,16)(17,18)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56730);