Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C5×M32 ⋊ C4 |
| Sylow | C5 ⋊ #2108 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C2×C4×C8 |
| Fitting | C5×M32 | C4 |
| prime radical | C22×D5 | C2×C8 |
| cosocle | C8×D5 | C23 |
| hypercenter | M32 | AGL(1,𝔽5) |
| hypocenter | C5 | #2108 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C8 ◁ M32 | |
184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 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=d,b^2=e,d^2=f,f^2=g,h^a=h^2,c^b=cg,h^b=h^3,h^d=h^4,h^e=h^4100 conjugacy classes — view the full character table.
| maximal subgroup of | #13025419, #13025748, #13025754, … |
| maximal normal subgroup of | #13025419, #13025748, #13025754, … |
| maximal quotient of | #13022967, #13022968, #13022978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 6195);