Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×D32×C8 |
| Fitting | C2×C32×C8 ⋊ C22 |
| derived, Sylow | C32 ⋊ C23×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | D32 |
| commutator, hypocenter | C32 | C23×C8 |
| Frattini | C4 | C22×D32 |
| Fitting | C2×C32×C8 | C22 |
| socle | C22×C32 | C22×C4 |
| prime radical | C22×D32 | C4 |
| cosocle | C32×C4 | C24 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C8 | |
714 conjugacy classes of subgroups; only the 256 normal subgroups are shown.
This group has 714 conjugacy classes of subgroups — too many to draw. Only its 256 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D32⋊C4, #11783692, #11785358, … |
| maximal normal subgroup of | C8×D32⋊C4, #11783692, #11785358, … |
| maximal quotient of | C4×D32×C8, C2×D3×C8×Dic3, C2×C8×#624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 6546);