Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C2×#63700 |
| Sylow | C2×H(5,𝔽2) ⋊ 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×H(5,𝔽2) | C3 |
| cosocle | C3×Q8 | C23×C3 |
| hypercenter | C2×C3×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×Q8 | |
381 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
114 conjugacy classes — view the full character table.
| maximal subgroup of | #2888⋊C9, #2897⋊C9, C4×#63700, … |
| maximal normal subgroup of | #2888⋊C9, #2897⋊C9, C4×#63700, … |
| maximal quotient of | C2×#520⋊C9, C4×#63700, C2×C4⋊(#41⋊C9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5000);