  We discuss a practical algorithm to compute parabolic Kazhdan-Lusztig
polynomials. As an application we compute Kazhdan-Lusztig polynomials which are
needed to evaluate a character formula for reductive groups due to Lusztig.
  Some coefficients of these polynomials have interesting interpretations for
certain finite groups. We find examples of finite dimensional modules for
finite groups with much higher dimensional first cohomology group than in all
previously known cases.
  Some of these examples lead to the construction of finite groups with many
maximal subgroups, contradicting an old conjecture by G.~E.~Wall.
