Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D5×#453 |
| Fitting | C5×#453 ⋊ C2 |
| Sylow | C5 ⋊ C2×#453 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5×#120 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#453 | C2 |
| socle | C2×C5 | C2×#120 |
| hypercenter | #453 | D5 |
| hypocenter | C5 | C2×#453 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #453 | |
844 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 844 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 40 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1402, #12019521, D5×#2196, … |
| maximal normal subgroup of | D5×#1402, #12019521, D5×#2196, … |
| maximal quotient of | D5×#1595, D5×#1596, D5×#1613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 14089);