Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C5×#131 ⋊ C4 |
| Sylow | C5 ⋊ #2875 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C23×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C22×C42 |
| Fitting | C5×#131 | C4 |
| prime radical | D5×#41 | C22 |
| cosocle | C4×D5 | C24 |
| hypercenter | #131 | AGL(1,𝔽5) |
| hypocenter | C5 | #2875 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ #131 | |
556 conjugacy classes of subgroups.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=f,e^2=g,b^a=bg,h^b=h^2,d^c=dg,h^f=h^488 conjugacy classes — view the full character table.
| maximal subgroup of | #13025725, #13025726, #13033815, … |
| maximal normal subgroup of | #13025725, #13025726, #13033815, … |
| maximal quotient of | #13028974, #13028975, #13028976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19497);