Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D5×#576 |
| Fitting | C5×#576 ⋊ C2 |
| Sylow | C5 ⋊ C2×#576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D4×D5 |
| commutator, cosocle | C4×C5 | C25 |
| Frattini | C4 | C24×D5 |
| Fitting | C5×#576 | C2 |
| socle | C2×C5 | C23×D4 |
| hypercenter | #576 | D5 |
| hypocenter | C5 | C2×#576 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #576 | |
1512 conjugacy classes of subgroups.
| transitive degree | μt(G) = 80 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#2192, D5×#2194, #12253262, … |
| maximal normal subgroup of | D5×#2192, D5×#2194, #12253262, … |
| maximal quotient of | D5×#2897, D5×#2913, D5×#2946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21185);