Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7106 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #331 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C22 | #1270 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 656 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) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4)(5,12,13,18,14,17,15,8)(6,9)(7,11)(10,16);(3,5)(4,8)(6,15)(7,14)(9,17)(10,12)(11,18)(13,16)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31251);