Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#7361 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #423 | C23 |
| socle | C22 | #1319 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
434 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 434 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);(3,4,6,10,8,12,15,17,9,13,16,18,7,11,5,14);(3,5)(4,10,13,18)(6,8)(7,16)(9,15)(11,14,12,17)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56787);