Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C64 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Sylow | C5 ⋊ C4×C64 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C4×C64 |
| Frattini | C2×C16 | C2×AGL(1,𝔽5) |
| Fitting | C2×C5×C32 | C4 |
| socle, prime radical | C22×C5 | C2×C32 |
| cosocle | C2×#3926 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C32 | |
86 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1280 |
Polycyclic presentation:
2^8,5|a^2=c,b^2=d,c^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, 1084786);