Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| Fitting | C5×#486 ⋊ C2 |
| Sylow | C5 ⋊ #1721 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #4086 |
| commutator, cosocle | C42×C5 | C23 |
| Frattini | C42 | C22×D5 |
| Fitting | C5×#486 | C2 |
| socle | C22×C5 | #127 |
| hypercenter | #486 | D5 |
| hypocenter | C5 | #1721 |
| LCS | G ▷ C42×C5 ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #486 | |
242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|b^2=d,c^2=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^4,c^b=cf,h^b=h^470 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1721, C2×#10601987, #12087091, … |
| maximal normal subgroup of | D5×#1721, C2×#10601987, #12087091, … |
| maximal quotient of | #11993091, #11999102, #11999103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 4069);