Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C4 |
| direct product | AGL(1,𝔽5)×C32 |
| Fitting | C5×C32 ⋊ C4 |
| derived, Sylow | C5 ⋊ C4×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C4×C32 |
| Frattini | C16 | C2×AGL(1,𝔽5) |
| Fitting | C5×C32 | C4 |
| socle | C2×C5 | C4×C16 |
| prime radical | C2×D5 | C2×C16 |
| cosocle | D5×C16 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C32 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×C64, #13014519, C2×AGL(1,𝔽5)×C32, … |
| maximal normal subgroup of | AGL(1,𝔽5)×C64, #13014519, C2×AGL(1,𝔽5)×C32, … |
| maximal quotient of | C32×#170, #170⋊C32, (C2×AGL(1,𝔽5))⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 567);