Groups with identical Charactertabels have a equal sized abilization

Question

I'd like to prove a question which i found on the internet unfortunatly without a solution.

Let G1 and G2 be two groups with the same character table. Show that |G1 : [G1, G1]| = |G2 : [G2, G2]|.

Where both Groups are assumed to be finte. Im hard stuck at this one, don't even no how to start i feel like I have to little information given. So the first question would be what one can deduce from knowing that two Groups have the same charactertabel, it can't be that it implies that they are Isomorphic, (consider $Q_{8}$ and $D_{8}$) Secondly i'd like to ask for some hints so i can work on the proof.

Answer 1

The number of one dimensional representations of a group equals the order of its abelianisation. This reult can be found here. The result then follows.

Answer 2

Even better, two finite groups with the same character table even have isomorphic abelianizations:

Does same character table imply isomorphic abelianizations?