Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#5966 |
| Fitting | C32×H(5,𝔽2) ⋊ C2 |
| Sylow | C32 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C24×D3 |
| commutator, cosocle | C2×C3 | C25×C3 |
| Frattini | C2 | C24×C3×D3 |
| Fitting | C32×H(5,𝔽2) | C2 |
| socle | C2×C32 | C25 |
| hypercenter | C3×H(5,𝔽2) | D3 |
| hypocenter | C3 | C3×#585 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C3×H(5,𝔽2) | |
1703 conjugacy classes of subgroups; only the 886 normal subgroups are shown.
This group has 1703 conjugacy classes of subgroups — too many to draw. Only its 886 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
153 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#80277, C3×#80291, #11864979, … |
| maximal normal subgroup of | C3×#80277, C3×#80291, #11864979, … |
| maximal quotient of | C3×#85856, C3×#85885, C3×#85857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8620);