Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C128 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C5 ⋊ C128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C32 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C128 |
| Fitting | C5×C32 | C4 |
| socle, prime radical | C2×C5 | C64 |
| cosocle | #64663 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C32 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^7,5|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^2,h^b=h^4160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10597921, D5⋊C128, #13014604, … |
| maximal normal subgroup of | C2×#10597921, D5⋊C128, #13014604, … |
| maximal quotient of | #13013205, C2×#10597921, C3×#10597921, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 3);