Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C5⋊#2104 |
| Sylow | C5 ⋊ #2104 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C5⋊#401 |
| commutator, prime radical, cosocle | C24×C5 | C23 |
| Frattini | C24 | C22×D5 |
| Fitting | C5×#342 | C2 |
| socle | C2×C5 | #401 |
| hypercenter | #342 | D5 |
| hypocenter | C5 | #2104 |
| LCS | G ▷ C24×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ #342 | |
178 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^7,5|a^2=d,b^2=df,c^2=de,b^a=bde,c^a=cef,e^a=eg,h^a=h^4,c^b=cf,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg67 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#2104, C2×C5⋊#2104, #12130289, … |
| maximal normal subgroup of | D5×#2104, C2×C5⋊#2104, #12130289, … |
| maximal quotient of | #12004441, #12004442, #12004443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 1825);