Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C2×2−1+4⋊C9 |
| unambiguous semidirect product | (C2×2−1+4)⋊C9 |
| Sylow | C2×2−1+4 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×A4 |
| commutator, hypocenter | Q8 | C23×C9 |
| Frattini | C2×C3 | C23×A4 |
| Fitting, prime radical | C2×C3×2−1+4 | C3 |
| cosocle | C3×Q8 | C23×C3 |
| hypercenter | C2×C3×D4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×D4 | |
365 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
114 conjugacy classes — view the full character table.
| maximal subgroup of | #2884⋊C9, #2889⋊C9, #2898⋊C9, … |
| maximal normal subgroup of | #2884⋊C9, #2889⋊C9, #2898⋊C9, … |
| maximal quotient of | C2×#519⋊C9, C2×#520⋊C9, C4×2−1+4⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5001);