Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C64 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C9⋊C64 |
| derived, Sylow | C9 ⋊ C64 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | D9 |
| commutator, hypocenter | C9 | C64 |
| Frattini | C3×C32 | D3 |
| Fitting, cosocle | C9×C32 | C2 |
| socle, prime radical | C2×C3 | C3⋊C32 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C32 | |
21 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^2|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,g^3=h^2,g^a=g^2h,h^a=h^2192 conjugacy classes — view the full character table.
| maximal subgroup of | D9×C64, C2×C9⋊C64, #11773094, … |
| maximal normal subgroup of | D9×C64, C2×C9⋊C64, #11773094, … |
| maximal quotient of | C9⋊C128, C2×C9⋊C64, C27⋊C64, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1);