Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32875 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2842 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2280 conjugacy classes of subgroups; only the 490 normal subgroups are shown.
This group has 2280 conjugacy classes of subgroups — too many to draw. Only its 490 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 — 5 generators on 18 points:
(1,2);(4,8)(5,11)(6,14)(9,16);(3,4,6,15,7,8,14,10)(5,12,13,9,11,17,18,16);(3,5,6,13,7,11,14,18)(4,9,10,12,8,16,15,17);(9,16)(12,17)56 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#32875⋊C3 |
| maximal subgroup of | #72268860, C2×#32875⋊C3 |
| maximal normal subgroup of | #72268860, C2×#32875⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538660);