First, for the purposes of this post I'll just call the center Z(G) as Z. It makes the notation easier to read.

So, G/Z(G) = G/Z is the set {gZ : g is in G}, where gZ = {gz : z is in Z}. This is the set of all left cosets of Z in G.

If a, b are elements of G, then the binary operation for G/Z takes any two elements aZ and bZ, and returns (ab)Z. In other words, (aZ)(bZ) = (ab)Z.