Which 58 does it hit?

Using a set of three PLL algs, I found that the most you can solve (with no more than 2 alg executions) is 71 out of 72 cases. (I'm basing the PLL names from the speedsolving.com Wiki page.) For example, with Ga, Ab, and Jb; you can solve all cases in 2 alg executions except Nb. So Ga, Ab, Jb, and Nb will solve all.

I calculated 16 cases where 71 out of 72 cases can be solved. Pick any G-Perm, any U- or A-Perm, and then some J- or R-Perm will allow you to reach 71 out of 72 cases with no more than 2 alg executions.

**So it appears to me that 4 is the smallest number of PLL algs that can be used to solve any PLL with at most 2 alg executions.**I calculated 16 cases where 71 out of 72 cases can be solved. Pick any G-Perm, any U- or A-Perm, and then some J- or R-Perm will allow you to reach 71 out of 72 cases with no more than 2 alg executions.