If two groups are generated by the same number of generators, and the generators have the same orders, are the groups isomorphic? If, say, for two groups G and H, G = < a1, a2, ... , a_n > and H = < b1, b2, ... , b_n >, 
such that |a_i| = |b_i| for all i from 1 to n, 
is G isomorphic to H? 
If so, what is the proof of this?
I know that for n = 1, then H and G are a cyclic group and are isomorphic to Z/|a|Z. Does this extend to larger generating sets?

HINT: (from @Daniel Fischer:)
Take the different groups $D_n$ of order $2n$ generated by two reflections (elements of order $2$) across two lines in the plane that form  an angle of $\frac{\pi}{n}$.
