Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C5×M32 ⋊ C2 |
| Sylow | C5 ⋊ #3407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C23×D5 |
| commutator, socle | C2×C5 | C23×C8 |
| Fitting | C2×C5×M32 | C2 |
| prime radical | C2×#709 | C4 |
| cosocle | C5×C8 | C24 |
| hypercenter | C2×M32 | D5 |
| hypocenter | C5 | #3407 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C8 ◁ C2×M32 | |
360 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=e,e^2=f,f^2=g,d^a=dg,c^b=cg,h^b=h^4148 conjugacy classes — view the full character table.
| maximal subgroup of | #12014424, #12014426, #12014427, … |
| maximal normal subgroup of | #12014424, #12014426, #12014427, … |
| maximal quotient of | #12251939, #12251940, #12251947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 15820);