Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C128 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D5⋊C128 |
| derived, Sylow | C5 ⋊ C2×C128 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C64 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C2×C128 |
| Frattini | C32 | C2×AGL(1,𝔽5) |
| Fitting | C5×C64 | C4 |
| socle | C2×C5 | C2×C64 |
| prime radical | C2×D5 | C64 |
| cosocle | #64663 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C64 | |
46 conjugacy classes of subgroups.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^8,5|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,g^2=h,i^a=i^2,i^b=i^4,i^c=i^4320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1084871);