  We provide a simple method to compute upper bounds on the essential dimension
of split reductive groups with finite or connected center by means of their
generically free representations. Combining our upper bound with previously
known lower bound, the exact value of the essential dimension is calculated for
some types of reductive groups. As an application, we determine the essential
dimension of a semisimple group of classical type or $E_{6}$, and its strict
reductive envelope under certain conditions on its center. This extends
previous works on simple simply connected groups of type $B$ or $D$ by
Brosnan-Reichstein-Vistoli and Chernousov-Merkurjev, strict reductive envelopes
of groups of type $A$ by Cernele-Reichstein, and semisimple groups of type $B$
by the authors to any classical type and type $E_{6}$ in a uniform way.
