Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C5×#564 ⋊ C4 |
| Sylow | C5 ⋊ #34530 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×AGL(1,𝔽5) |
| commutator, socle | C22×C5 | C24×C4 |
| Fitting | C5×#564 | C4 |
| prime radical, cosocle | C22×D5 | C25 |
| hypercenter | #564 | AGL(1,𝔽5) |
| hypocenter | C5 | #34530 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #564 | |
1800 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^8,5|a^2=h,b^2=g,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,i^a=i^3,c^b=cg,d^b=df,i^c=i^3,i^e=i^2,i^h=i^495 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1114641);