Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| isoclinism | isoclinic class of #66297 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | C24⋊C5 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #66297 |
| commutator, hypocenter | C24⋊C5 | C8 |
| Fitting, socle | C25 | AGL(1,𝔽5) |
| prime radical | C2×C24⋊C5 | C4 |
| cosocle | #66245 | C2 |
| LCS | G ▷ C24⋊C5 | |
| UCS | C1 ◁ C2 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 40 |
Polycyclic presentation:
2^3,5,2^4|a^2=b,b^2=c,d^a=d^2,e^a=h,f^a=fgh,g^a=egh,h^a=efg,d^b=d^4,e^b=efg,f^b=g,g^b=f,h^b=efh,e^d=efg,f^d=ef,g^d=gh,h^d=eg22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10619374, ((C24⋊C5)⋊C2)⋊C8, #13046100, … |
| maximal normal subgroup of | C2×#10619374, ((C24⋊C5)⋊C2)⋊C8, #13046100, … |
| maximal quotient of | #13046090, #13046091, #13046092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21456);