Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of C82⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C8 |
| unambiguous semidirect product | C82⋊C9 |
| derived, Sylow | C82 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C82⋊C3 |
| commutator, hypocenter | C82 | C9 |
| Frattini | C3×C42 | A4 |
| Fitting, cosocle | C3×C82 | C3 |
| socle, prime radical | C22×C3 | C42⋊C3 |
| LCS | G ▷ C82 | |
| UCS | C1 ◁ C3 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
3^2,2^6|a^3=b,c^2=ef,d^2=eh,e^2=gh,f^2=g,c^a=d,d^a=cdf,e^a=fgh,f^a=ef,g^a=h,h^a=gh72 conjugacy classes — view the full character table.
| derived subgroup of | #11919920 |
| maximal subgroup of | C2×C82⋊C9, #1454⋊C9, #1455⋊C9, … |
| maximal normal subgroup of | C2×C82⋊C9, #1454⋊C9, #1455⋊C9, … |
| maximal quotient of | #1273⋊C9, C2×C82⋊C9, C82⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 3);