  It is a classical problem to compute a minimal set of invariant polynomial
generating the invariant ring of a finite group as an algebra. We present here
an algorithm for the computation of minimal generating sets in the non-modular
case. Apart from very few explicit computations of Groebner bases, the
algorithm only involves very basic operations, and is thus rather fast.
  As a test bed for comparative benchmarks, we use transitive permutation
groups on 7 and 8 variables. In most examples, our algorithm implemented in
Singular works much faster than the one used in Magma, namely by factors
between 50 and 1000. We also compute some further examples on more than 8
variables, including a minimal generating set for the natural action of the
cyclic group of order 11 in characteristic 0 and of order 15 in characteristic
2.
  We also apply our algorithm to the computation of irreducible secondary
invariants.
