Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×C8×D9 |
| Fitting | C5×C8×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C2×C5×C8 |
| Hall | C5×C9 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C8 | D9 |
| commutator, hypocenter | C9 | C2×C5×C8 |
| Frattini | C3×C4 | C2×C5×D3 |
| Fitting | C5×C8×C9 | C2 |
| socle | C2×C3×C5 | C4×D3 |
| prime radical | C2×C5×D9 | C4 |
| cosocle | C4×C9 | C22×C5 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C5×C8 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#672, #13083692, C5×C16×D9, … |
| maximal normal subgroup of | D9×#672, #13083692, C5×C16×D9, … |
| maximal quotient of | C5×C16×D9, C5×#63357, C5×C8×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 50);