Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C5×#407 ⋊ C2 |
| Sylow | C5 ⋊ D4×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D5 |
| commutator, socle | C22×C5 | C23×C4 |
| Fitting | C5×#407 | C2 |
| prime radical | C22×C22⋊D5 | C4 |
| cosocle | C2×C4×C5 | C24 |
| hypercenter | #407 | D5 |
| hypocenter | C5 | D4×M16 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ #407 | |
544 conjugacy classes of subgroups.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,d^c=dg,h^c=h^4106 conjugacy classes — view the full character table.
| maximal subgroup of | #12114645, #12114647, #12114653, … |
| maximal normal subgroup of | #12114645, #12114647, #12114653, … |
| maximal quotient of | #12070718, #12070726, #12070734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 12441);