  In Fialowski's classification for algebras of maximal class, there are three
Lie algebras of maximal class with 1-dimensional homogeneous components:
$\mathfrak{m}_0$, $L_1$ and $\mathfrak{m}_2$. In this paper, we studied their
biderivations by considering the embedded mapping to derivation algebras. Then
we determined commuting mappings on these algebras as an application of
biderivations. Finally,
  local and 2-local derivations for these three algebras were characterized as
the given gradings.
