Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#5709 |
| Fitting | C32×#126 ⋊ C2 |
| Sylow | C32 ⋊ #553 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C23×D3 |
| commutator, cosocle | C22×C3 | C24×C3 |
| Frattini | C22 | C23×C3×D3 |
| Fitting | C32×#126 | C2 |
| socle | C22×C32 | C24 |
| hypercenter | C3×#126 | D3 |
| hypocenter | C3 | C3×#553 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#126 | |
506 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
117 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×#553, C2×C3×#5709, D3×#5709, … |
| maximal normal subgroup of | C3×D3×#553, C2×C3×#5709, D3×#5709, … |
| maximal quotient of | C3×#74962, C3×#74263, C3×#75305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7688);