Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C218 |
| direct product | C23×#379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C23 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C28 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
225122 conjugacy classes of subgroups; only the 14466 normal subgroups are shown.
This group has 225122 conjugacy classes of subgroups — too many to draw. Only its 14466 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) = 128 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4);(5,6);(8,10)(11,12,13,14)(16,18);(15,16)(17,18);(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532419);