Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | H(7,𝔽2)⋊C5 |
| Fitting, Sylow | H(7,𝔽2) ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C22×C24⋊C5 |
| commutator, cosocle, hypocenter | 2−1+4 | C22×C5 |
| Fitting | H(7,𝔽2) | C5 |
| hypercenter | Q8 | C24⋊C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
619 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 619 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^2,5,2^5|a^2=h,b^2=h,e^2=h,b^a=bh,d^c=gh,e^c=dg,f^c=de,g^c=efgh,e^d=eh,f^d=fh,g^d=gh,g^f=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #63091⋊C5, #63092⋊C5, C2×H(7,𝔽2)⋊C5, … |
| maximal normal subgroup of | #63091⋊C5, #63092⋊C5, C2×H(7,𝔽2)⋊C5, … |
| maximal quotient of | #62654⋊C5, Q8×2−1+4⋊C5, C4⋊(#585⋊C5), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21473);