Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C22⋊((C24⋊C5)⋊C2) |
| Fitting | C26 ⋊ D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×(C24⋊C5)⋊C2 |
| commutator, cosocle | C2×C24⋊C5 | C22 |
| Fitting | C26 | D5 |
| socle | C25 | C2×D5 |
| hypercenter | C22 | (C24⋊C5)⋊C2 |
| hypocenter | C24⋊C5 | D4 |
| LCS | G ▷ C2×C24⋊C5 ▷ C24⋊C5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
601 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 601 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 20 |
Polycyclic presentation:
2^3,5,2^4|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 | D4×(C24⋊C5)⋊C2, #13046175, C2×C22⋊((C24⋊C5)⋊C2), … |
| maximal normal subgroup of | D4×(C24⋊C5)⋊C2, #13046175, C2×C22⋊((C24⋊C5)⋊C2), … |
| maximal quotient of | C22⋊#66245, #13046118, #13046119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21467);