Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C43 |
| direct product | C2×#33300 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5514 conjugacy classes of subgroups; only the 1330 normal subgroups are shown.
This group has 5514 conjugacy classes of subgroups — too many to draw. Only its 1330 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) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,6)(8,10)(12,15)(13,17)(16,18);(11,12,13,15,14,16,17,18);(3,4,5,6);(7,8,9,10)152 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#33300⋊C3 |
| maximal subgroup of | C2×#33300⋊C3, #72340569 |
| maximal normal subgroup of | C2×#33300⋊C3, #72340569 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538131);