Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D9×#135 |
| Fitting | C9×#135 ⋊ C2 |
| Sylow | C9 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D9 |
| commutator, cosocle | C4×C9 | C24 |
| Frattini | C3×C4 | C23×D3 |
| Fitting | C9×#135 | C2 |
| socle | C2×C3 | C2×D3×D4 |
| hypercenter | #135 | D9 |
| hypocenter | C9 | C2×#135 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
399 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 399 conjugacy classes of subgroups — too many to draw. Only its 121 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×#359, #11796781, D9×#360, … |
| maximal normal subgroup of | D9×#359, #11796781, D9×#360, … |
| maximal quotient of | D9×#416, D9×#427, C4×D8×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1751);