Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×#3710 |
| derived, Sylow | C32 ⋊ C2×C5×C8 |
| Hall | C32×C5 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | D32 |
| commutator, hypocenter | C32 | C2×C5×C8 |
| Frattini | C4 | C5×D32 |
| Fitting | C32×C4×C5 | C22 |
| socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C5×C3⋊D3 | C4 |
| cosocle | C32×C4 | C22×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4×C5 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D32⋊C8, C5×#63729, C5×#63730, … |
| maximal normal subgroup of | C5×D32⋊C8, C5×#63729, C5×#63730, … |
| maximal quotient of | C5×#63542, C5×#63544, C5×#63553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 246);