Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×C5×D8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32×C5 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C32×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C5 ◁ C32×C4×C5 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(7,8,9,10,11)(12,13,14,15,16,17,18,19);(4,5,6)(13,19)(14,18)(15,17)315 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 388);