K-theoretic invariants for unital Kirchberg algebras with finitely generated K-groups

We introduce a hierarchy for unital Kirchberg algebras with finitely generated K-groups by which the 1st and 2nd homotopy groups of the automorphism groups serve as a complete invariant of classification. We also give a complete invariant specific to the case of unital Kirchberg algebras with finitely generated K-groups and provide a useful tool to classify the Cuntz--Krieger algebras.