Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×C4×#674 |
| derived, Sylow | C5 ⋊ C2×C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C2×C4×C16 |
| Frattini | C2×C4 | C22×AGL(1,𝔽5) |
| Fitting | C2×C42×C5 | C4 |
| socle, prime radical | C23×C5 | C2×C8 |
| cosocle | C2×#168 | C23 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C42 | |
216 conjugacy classes of subgroups.
| transitive degree | μt(G) = 640 |
160 conjugacy classes — view the full character table.
| maximal subgroup of | #13013464, #13013475, #13013543, … |
| maximal normal subgroup of | #13013464, #13013475, #13013543, … |
| maximal quotient of | C2×C8×#674, C2×#10598513, M16×#674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 6228);