Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C5 ⋊ #3407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C23×C8 |
| Fitting | C5×#131 | C4 |
| prime radical | D5×#41 | C4 |
| cosocle | #168 | C24 |
| hypercenter | #131 | AGL(1,𝔽5) |
| hypocenter | C5 | #3407 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ #131 | |
360 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=e,e^2=f,f^2=g,d^a=dg,h^a=h^2,c^b=cg,h^d=h^4,h^e=h^488 conjugacy classes — view the full character table.
| maximal subgroup of | #13025782, #13025783, #13026940, … |
| maximal normal subgroup of | #13025782, #13025783, #13026940, … |
| maximal quotient of | #13043111, #13043122, #13043123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19495);