Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C2×#585⋊C5 |
| unambiguous semidirect product | (C2×#585)⋊C5 |
| Fitting, Sylow | C2×#585 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C24⋊C5 |
| commutator, cosocle, hypocenter | 2−1+4 | C22×C5 |
| Frattini | C2 | C22×C24⋊C5 |
| Fitting | C2×#585 | C5 |
| socle | C22 | C2×C24⋊C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C2×C4 | |
619 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 619 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#585⋊C5, #62654⋊C5, C22⋊(#585⋊C5), … |
| maximal normal subgroup of | C4×#585⋊C5, #62654⋊C5, C22⋊(#585⋊C5), … |
| maximal quotient of | C2×C4×2−1+4⋊C5, C4×#585⋊C5, C2×#3531⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21472);