Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C25×AGL(1,𝔽5) |
| unambiguous semidirect product | D5⋊(C24×C4) |
| Fitting, socle | C25×C5 ⋊ C4 |
| derived, Sylow | C5 ⋊ C25×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C25×C4 |
| Frattini | C1 | G |
| Fitting, socle | C25×C5 | C4 |
| prime radical | C25×D5 | C2 |
| cosocle | D5 | C26 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C25 | |
10552 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21539);