Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#63710 |
| Fitting | C22×D4×C9 ⋊ C2 |
| Sylow | C9 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D9 |
| commutator, cosocle | C2×C9 | C25 |
| Frattini | C2×C3 | C24×D3 |
| Fitting | C22×D4×C9 | C2 |
| socle | C22×C3 | C23×D3 |
| hypercenter | C22×D4 | D9 |
| hypocenter | C9 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
1347 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 1347 conjugacy classes of subgroups — too many to draw. Only its 517 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
114 conjugacy classes — view the full character table.
| maximal subgroup of | #11784811, #11784814, #11784852, … |
| maximal normal subgroup of | #11784811, #11784814, #11784852, … |
| maximal quotient of | C2×#10589647, C2×#10589588, C2×#10589900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5014);