Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| direct product | C2×#66244 |
| Fitting | C2×2−1+4 ⋊ D5 |
| derived | 2−1+4⋊C5 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | (C24⋊C5)⋊C2 |
| commutator, cosocle, hypocenter | 2−1+4⋊C5 | C22 |
| Frattini | C2 | C2×(C24⋊C5)⋊C2 |
| Fitting | C2×2−1+4 | D5 |
| LCS | G ▷ 2−1+4⋊C5 | |
| UCS | C1 ◁ C22 | |
229 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10619373, #13046109, C4×#66244, … |
| maximal normal subgroup of | C2×#10619373, #13046109, C4×#66244, … |
| maximal quotient of | C2×#10617014, C2×#10617013, #13046122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21461);