Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8 · Schur multiplier (H2) = C43 |
| direct product | C42×#170 |
| derived, Sylow | C5 ⋊ C42×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C42×C8 |
| Frattini | C23 | C22×AGL(1,𝔽5) |
| Fitting | C2×C42×C5 | C4 |
| socle, prime radical | C23×C5 | C22×C4 |
| cosocle | C22×Dic5 | C23 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C42 | |
408 conjugacy classes of subgroups.
| transitive degree | μt(G) = 640 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | #13013207, C4×C8×#170, C4×#64879, … |
| maximal normal subgroup of | #13013207, C4×C8×#170, C4×#64879, … |
| maximal quotient of | C4×C8×#170, C4×#64879, #97×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 6251);