Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#64308 |
| Fitting | C22×C32×C4 ⋊ C22 |
| Sylow | C32 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D32 |
| commutator, cosocle | C2×C32 | C25 |
| Frattini | C2 | C23×D32 |
| Fitting | C22×C32×C4 | C22 |
| socle | C22×C32 | C24 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
1914 conjugacy classes of subgroups; only the 536 normal subgroups are shown.
This group has 1914 conjugacy classes of subgroups — too many to draw. Only its 536 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | #11784829, #11784866, #11784874, … |
| maximal normal subgroup of | #11784829, #11784866, #11784874, … |
| maximal quotient of | C2×#10595058, C2×#10594996, C2×#10593994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8550);