Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | Q8×C24⋊C5 |
| unambiguous semidirect product | (C24×Q8)⋊C5 |
| Fitting, Sylow | C24×Q8 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C24⋊C5 |
| commutator, socle, cosocle | C25 | C22×C5 |
| Fitting | C24×Q8 | C5 |
| prime radical | C2×C24⋊C5 | C22 |
| hypercenter | Q8 | C24⋊C5 |
| hypocenter | C24 | C5×Q8 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
648 conjugacy classes of subgroups.
| transitive degree | μt(G) = 40 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×C24⋊C5, Q16×C24⋊C5, C2×Q8×C24⋊C5, … |
| maximal normal subgroup of | SD16×C24⋊C5, Q16×C24⋊C5, C2×Q8×C24⋊C5, … |
| maximal quotient of | C4⋊C4×C24⋊C5, Q8×2−1+4⋊C5, #63004⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21470);