Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | D5×#585 |
| Fitting | C5×#585 ⋊ C2 |
| Sylow | C5 ⋊ C2×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24×D5 |
| commutator, socle, cosocle | C2×C5 | C26 |
| Frattini | C2 | C25×D5 |
| Fitting | C5×#585 | C2 |
| hypercenter | #585 | D5 |
| hypocenter | C5 | C2×#585 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ #585 | |
5956 conjugacy classes of subgroups; only the 3203 normal subgroups are shown.
This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
136 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1791, #12098577, D5×#1794, … |
| maximal normal subgroup of | D5×#1791, #12098577, D5×#1794, … |
| maximal quotient of | D5×#3428, C4×D5×H(5,𝔽2), C4×D5×2−1+4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21523);