Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #504⋊C9 |
| Sylow | #504 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C16 | A4 |
| commutator, hypocenter | Q8 | C8×C9 |
| Frattini | C3×C8 | C2×A4 |
| Fitting | C3×#504 | C3 |
| socle | C2×C3 | C8×A4 |
| prime radical | C3×#41 | C3×C4 |
| cosocle | C3×#131 | C2×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C3×C16 | |
49 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3,2,3,2^4|a^2=c,b^3=d,c^2=e,e^2=h,f^2=h,g^2=h,f^b=g,g^b=fg,g^f=gh168 conjugacy classes — view the full character table.
| maximal subgroup of | #2258⋊C9, C2×#504⋊C9, #3407⋊C9, … |
| maximal normal subgroup of | #2258⋊C9, C2×#504⋊C9, #3407⋊C9, … |
| maximal quotient of | C16×Q8⋊C9, C2×#504⋊C9, #504⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 207);