Generating systems of $S_7$

Question

My exercise is the following:

$i)$ Is {$(5 1), (7134256)$} a generating system of $S_7$?

$ii)$ Is {$(521), (7134256)$} a generating system of $S_7$?

In the lecture we wrote down the following:

The following sets are generating systems of $S_n$:

$1)$ {$(12),(23),...,(n-1 n)$}

$2)$ {$(12),(12...n)$}

So with $i)$ I just showed that you can combine $(5 1), (7134256)$ to get $(12)$ and $(12...n)$, which is a generating system according to the above. My problem is the second one: I cannot manage to show that one could combine $(521),(7134256)$ to get $(12)$. So my assumption right now is that the set given in $ii)$ is not a generating system. But that also means that I have to show why it can't be one (since the theorem doesn't imply that there are no other types of generating systems for $S_n$). I can't just write out all elements, obviously. But how else can I show that it's not a generating system? (If it is in fact one - how can I show that?)

Thanks in advance.

Answer 1

What is the sign of an element in the subgroup generated by $\{(521), (7134256)\}$?