Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C4⋊((C24⋊C5)⋊C2) |
| Fitting | C24×C4 ⋊ D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×(C24⋊C5)⋊C2 |
| commutator, cosocle | C2×C24⋊C5 | C22 |
| Fitting | C24×C4 | D5 |
| socle | C25 | C2×D5 |
| hypercenter | C4 | (C24⋊C5)⋊C2 |
| hypocenter | C24⋊C5 | D4 |
| LCS | G ▷ C2×C24⋊C5 ▷ C24⋊C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
358 conjugacy classes of subgroups.
| transitive degree | μt(G) = 20 |
Polycyclic presentation:
2^3,5,2^4|b^2=c,b^a=bc,d^a=d^4,e^a=h,g^a=efgh,h^a=e,e^d=eg,f^d=ef,g^d=fgh,h^d=e28 conjugacy classes — view the full character table.
| maximal subgroup of | #13046147, #13046148, #13046162, … |
| maximal normal subgroup of | #13046147, #13046148, #13046162, … |
| maximal quotient of | C4⋊#66245, #13046119, C4⋊#66244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21459);