Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#65844 |
| Fitting | C22×C5×C4⋊C4 ⋊ C2 |
| Sylow | C5 ⋊ C2×#537 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D5 |
| commutator, cosocle | C22×C5 | C25 |
| Frattini | C22 | C24×D5 |
| Fitting | C22×C5×C4⋊C4 | C2 |
| socle | C23×C5 | C24 |
| hypercenter | C22×C4⋊C4 | D5 |
| hypocenter | C5 | C2×#537 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C22×C4⋊C4 | |
1640 conjugacy classes of subgroups; only the 565 normal subgroups are shown.
This group has 1640 conjugacy classes of subgroups — too many to draw. Only its 565 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
124 conjugacy classes — view the full character table.
| maximal subgroup of | #11991106, #11991109, #12005885, … |
| maximal normal subgroup of | #11991106, #11991109, #12005885, … |
| maximal quotient of | C2×#10613917, C2×#10613887, C2×#10605493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19697);