Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D9×#123 |
| Fitting | C9×#123 ⋊ C2 |
| Sylow | C9 ⋊ C2×#123 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D9 |
| commutator, cosocle | C22×C9 | C24 |
| Frattini, socle | C22×C3 | C23×D3 |
| Fitting | C9×#123 | C2 |
| prime radical | C2×D4×D9 | C2 |
| hypercenter | #123 | D9 |
| hypocenter | C9 | C2×#123 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ #123 | |
513 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#352, D9×#458, C2×D9×#123, … |
| maximal normal subgroup of | D9×#352, D9×#458, C2×D9×#123, … |
| maximal quotient of | D9×#381, D9×#385, D9×#386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1636);