Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32×C5 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×C5×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32×C5 | C24 |
| commutator, Frattini, cosocle | C2 | C24×C32×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C5 ◁ G | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(7,8,9,10,11)(12,13)(14,18)(15,16)(17,19);(4,5,6)(13,16)(17,18);(12,13)(14,17)(15,16)(18,19);(12,14)(13,17)(15,19)(16,18)765 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5839);