Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C16 · Schur multiplier (H2) = C1 |
| direct product | C5×#3771 |
| derived, Sylow | C32 ⋊ C5×C16 |
| Hall | C32×C5 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C5×C16 |
| Frattini | C2 | C5×AGL(1,𝔽9) |
| Fitting, socle, prime radical | C2×C32×C5 | C8 |
| cosocle | #622 | C2×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C5 | |
26 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#3771, #13088387, #13088388, … |
| maximal normal subgroup of | D5×#3771, #13088387, #13088388, … |
| maximal quotient of | C5×#63725, C2×C5×#3771 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 422);