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×C5×2−1+4 |
| 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 | |
876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(7,8,9,10,11)(13,17)(15,23)(18,20)(21,27);(4,5,6)(12,13,16,17)(14,20,22,18)(15,19,23,24)(21,26,27,25);(12,14,16,22)(13,18,17,20)(15,21,23,27)(19,25,24,26);(12,15)(13,19)(14,21)(16,23)(17,24)(18,25)(20,26)(22,27)765 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5840);