Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#64320 |
| Fitting | C32×C5×Q8 ⋊ C22 |
| Hall | C32×C5 ⋊ H(5,𝔽2) |
| Sylow | C32 ⋊ C5×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C22×D32 |
| commutator, cosocle | C2×C32 | C24×C5 |
| Frattini | C2 | C22×C5×D32 |
| Fitting | C32×C5×Q8 | C22 |
| socle | C2×C32×C5 | C24 |
| hypercenter | C5×Q8 | D32 |
| hypocenter | C32 | C5×H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C5 ◁ C5×Q8 | |
704 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3,4,5)(6,8,9,12)(7,11,10,13)(14,15,16)(17,18,19);(6,7,9,10)(8,13,12,11);(7,10)(11,13)(15,16);(8,12)(11,13)(18,19)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5650);