Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×AGL(1,𝔽9) |
| Fitting, Sylow | C32×C9 ⋊ C8 |
| derived | C32 ⋊ C8×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C8×C9 |
| Frattini | C3 | C3×AGL(1,𝔽9) |
| Fitting | C32×C9 | C8 |
| socle | C33 | C3×C8 |
| prime radical | C3×C3⋊D3 | C3×C4 |
| cosocle | C3×#157 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C9 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
81 conjugacy classes — view the full character table.
| maximal subgroup of | D9×AGL(1,𝔽9), C9×AΓL(1,𝔽9), AGL(1,𝔽9)⋊D9, … |
| maximal normal subgroup of | D9×AGL(1,𝔽9), C9×AΓL(1,𝔽9), AGL(1,𝔽9)⋊D9, … |
| maximal quotient of | C9×#3771, C2×C9×AGL(1,𝔽9), C9×#6249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 535);