Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#5647 |
| Fitting | C32×C22⋊Q8 ⋊ C2 |
| Sylow | C32 ⋊ #562 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C23×D3 |
| commutator, cosocle | C22×C3 | C24×C3 |
| Frattini | C22 | C23×C3×D3 |
| Fitting | C32×C22⋊Q8 | C2 |
| socle | C22×C32 | C24 |
| hypercenter | C3×C22⋊Q8 | D3 |
| hypocenter | C3 | C3×#562 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C22⋊Q8 | |
477 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
99 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×#562, C2×C3×#5647, D3×#5647, … |
| maximal normal subgroup of | C3×D3×#562, C2×C3×#5647, D3×#5647, … |
| maximal quotient of | C3×#74013, C3×#76207, C3×#75312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 7626);