Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C5×#137 ⋊ C2 |
| Sylow | C5 ⋊ #3586 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D4×D5 |
| commutator, cosocle | C4×C5 | C25 |
| Frattini | C4 | C24×D5 |
| Fitting | C2×C5×#137 | C2 |
| socle | C2×C5 | C23×D4 |
| hypercenter | C2×#137 | D5 |
| hypocenter | C5 | #3586 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#137 | |
1412 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|b^2=g,f^2=g,b^a=bf,e^a=eg,f^a=fg,f^b=fg,d^c=dg,h^c=h^485 conjugacy classes — view the full character table.
| maximal subgroup of | #12130598, #12130600, #12130620, … |
| maximal normal subgroup of | #12130598, #12130600, #12130620, … |
| maximal quotient of | #12962874, #12962880, #12962981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 21180);