An independent set of a graph \( G \) is a set of vertices of \( G \) which are pairwise non-adjacent. There are many applications for which the input is a graph \( G \) with a large symmetry group and the goal is to generate up to isomorphism all of the independent sets, all of the maximal independent sets, or all of the maximum independent sets. This paper presents a very fast practical algorithm for these problems. The tactic can also be applied to many other problems: some examples are generation of all dominating sets, colorings, or matchings of a graph up to isomorphism.