  We compute generators for the Chow ring of the classifying space of PGL_3
(over the complex numbers) as defined by Totaro. We also find enough relations
after inverting 3. We show that this ring is not generated by Chern classes
(this is the first example of this kind among classical groups) and prove that
Totaro's refined cycle class map to a quotient of complex cobordism of BPGL_3
is surjective
