Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| direct product | D5×Q64 |
| Fitting | C5×Q64 ⋊ C2 |
| Sylow | C5 ⋊ C2×Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5×D16 |
| commutator, cosocle | C5×C16 | C23 |
| Frattini | C16 | C22×D5 |
| Fitting | C5×Q64 | C2 |
| socle | C2×C5 | C2×D16 |
| prime radical | C2×D5 | D16 |
| hypercenter | Q64 | D5 |
| hypocenter | C5 | C2×Q64 |
| LCS | G ▷ C5×C16 ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q64 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
76 conjugacy classes — view the full character table.
| maximal subgroup of | D5×SD128, D5×Q128, #12020361, … |
| maximal normal subgroup of | D5×SD128, D5×Q128, #12020361, … |
| maximal quotient of | D5×#1416, D5×#1423, #12019883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 2257);