Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | D5×#496 |
| Fitting | C5×#496 ⋊ C2 |
| Sylow | C5 ⋊ C2×#496 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D4×D5 |
| commutator, cosocle | C2×C4×C5 | C24 |
| Frattini | C2×C4 | C23×D5 |
| Fitting | C5×#496 | C2 |
| socle | C22×C5 | C22×D4 |
| hypercenter | #496 | D5 |
| hypocenter | C5 | C2×#496 |
| LCS | G ▷ C2×C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #496 | |
668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#1406, D5×#1655, D5×#1656, … |
| maximal normal subgroup of | D5×#1406, D5×#1655, D5×#1656, … |
| maximal quotient of | D5×#1714, D5×#1715, D5×#1717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 15656);