Heading: A General Inquiry about Finite Isomorphic Groups and Their Generators
If I am given two finite groups whose generators I know, is it enough to show that by mapping the generators appropriately, and showing that the generators behave the same with respect to their group operation, the two groups are isomorphic?
I am inclined to say the answer is yes. I used this example to motivate my understanding:$$\langle(56),(1234)\rangle \cong \{1,3,9,11,13,17,19\}=\langle{3,11}\rangle$$where the operation on the group to the right is multiplication mod(20). I used the following map:$$\varphi :(56)\rightarrow11,\space \varphi:(1234)\rightarrow3$$ Everything maps out nicely when I write the multiplication tables for each group.
Yes, if you define "behave the same" in a suitable way. It is possible to write down a finite set of identities in the generators that have to hold (in your case they would be $a^2, b^4, ab=ba$ for $a=(5,6)$ and $b=(1,2,3,4)$) such that any map on the generators, whose images fulfill these identities, is a homomorphism. Such a set of identities is called a presentation, but it is not always obvious (or easy to determine) what a presentation in given generators would have to be.
Similarly one can write identities (expressions for elements in nontrivial normal subgroups) that would have to evaluate nontrivially in the images for a homomorphism for it to be an isomorphism. (In your example these would be $a^2$, $b^2$ and $ab^2$.)