Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C5×#421 ⋊ C2 |
| Sylow | C5 ⋊ #2008 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #65249 |
| commutator, cosocle | C22×C4×C5 | C23 |
| Frattini | C22×C4 | C22×D5 |
| Fitting | C5×#421 | C2 |
| socle | C2×C5 | #392 |
| hypercenter | #421 | D5 |
| hypocenter | C5 | #2008 |
| LCS | G ▷ C22×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #421 | |
380 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^7,5|d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^4,c^b=cf,d^b=dg,d^c=dg,e^c=eg,h^c=h^467 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#2008, C2×#10601202, #12126307, … |
| maximal normal subgroup of | D5×#2008, C2×#10601202, #12126307, … |
| maximal quotient of | #11999609, #11999623, #11999637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 3284);