Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D9×H(5,𝔽2) |
| Fitting | C9×H(5,𝔽2) ⋊ C2 |
| Sylow | C9 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×D9 |
| commutator, cosocle | C2×C9 | C25 |
| Frattini, socle | C2×C3 | C24×D3 |
| Fitting | C9×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | D9 |
| hypocenter | C9 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
1347 conjugacy classes of subgroups; only the 513 normal subgroups are shown.
This group has 1347 conjugacy classes of subgroups — too many to draw. Only its 513 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
102 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#453, D9×#455, #11865015, … |
| maximal normal subgroup of | D9×#453, D9×#455, #11865015, … |
| maximal quotient of | D9×#518, D9×#520, #11900157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5022);