EmbeddingsofPSL(2, 41) andPSL(2, 49) inE8(C)

We classify embeddings of the finite simple groups PSL(2, 41) andPSL (2, 49) into algebraic groups of type E8in characteristic not dividing the orders of these respective groups. Lie theory and finite group theory point to families of elements in a group of typeE8 over a finite field which might satisfy a presentation for our finite simple group. Techniques from computational representation theory identify all suitable systems of generators. We describe a number of methods of computation in Chevalley groups of type E8.