A Pentium would be a 686. 586 processors were intermediate processors made by Cyrix, AMD, and other (often they fit in 486 sockets).
You mean the
Am5x86 and the
Cx5x86, all aimed at 486 motherboards.
Nexgen also had the
Nx586, but this had its own socket and was more or less comparable to a Pentium I cpu.
The 586 is indeed the Pentium I (with or without MMX). Pentium Pro is, as mentioned in another post, a 686.
