Presentations of $D_8$ using permutations

Question

I fond a lot of examples using presentation of $D_8$ by generators which are permutations of $S_4$.

1) How many presentations could be found?

2) Could it be presented by permutations of $S_5$ or any different group?

Thanks for advance

Answer 1

It is not entirely clear what the OP is asking.However reading the comments I suspect what they mean may be along the lines of how many subsets of $D_{8}$ generate the full group. I will provide an answer here so the question does not remain listed as unanswered forever...

There are 255 non empty subsets of the elements of $D_8$. Looking at the groups these subsets generate, 216 of them generate all of $D_{8}$.

The OPs second question seems to be about realising $D_8$ as a subgroup of a permutation group. They assert that you can generate $D_8$ with permutations from $S_4$. Indeed this is true, for instance $\langle (3,4), (1,4)(2,3) \rangle $. They then seem to ask can we do the same thing with permutations from $S_{5}$. First, we could just see these elements as living in $S_{5}$ instead of $S_{4}$. However perhaps what the OP means is something like $\langle (3,5), (1,5)(2,3) \rangle $.

To the OP, if this has not suitably addressed your question, please update the question with more clarification.