I'm not sure, but the answer seems to be $2 {{n} \choose 2}$ Just rotate the $l$, then whenever it becomes perpendicular with some line $P_i P_j$, i and j switches their position on the permutation. When $l$ is rotating, it becomes perpendicular with $P_i P_j$ 2 times for specific $i, j$, so new permutation occurs $2 {{n} \choose 2}$ times in total.

For two parallel and directed lines with the same direction, the order of projections of $A_1 , A_2 \dots A_8$ must be the same. So, we only need to look at all the directed lines passing through a fixed point, say $O$ If a directed line is taken perpendicular to a line joining two given points, then the projections of these two points must coincide, and a corresponding permutation will not be produced. When the directed lines taken are not perpendicular to any line joining two given points, any two projections of $A_1 , A_2 , \dots, A_8$ must not coincide. There is therefore a corresponding permutation. (1) Suppose that the number of lines through point $O$ that are also perpendicular to a line joining two given points is $k$. Then $k\leq \dbinom{8}{2}= 28$. Then there arise $2k$ directed lines placed anticlockwise. Assume that they are in order of $l_1 , l_2 , \dots, l_{2k}$. For an arbitrary directed line $l$ (different from $l_1 , \dots, l_{2k}$ ) , there must be two consecutive directed lines $l_j$ and $l_{j+ 1}$ such that $l_j , l , l_{j+ 1}$ are placed anticlockwise. We can see that for a given $j$, the corresponding permutations obtained from such $l$ must be the same. (2)For any two directed lines $l$ and $l'$ different from $l_1, \dots l_{2k}$, if we cannot find $j$ such that both $l_j, l, l_{j+1}$ and $l_j, l', l_j+1$ satisfy (1) then there must be $l$ so that $l’, l_j , l$ are placed anticlockwise. Assume that $l_j$ is perpendicular to the line joining $A_{j_1}$, and $A_{j_2}$; the orders of the projections of points $A_{j_1}$, and $A_{j_2}$ on the directed lines $l$ and $l’$ must be different, so the corresponding permutations must also be different. It follows from (1) and (2) that the number of different permutations is $2k$. Note that $k=\dbinom{8}{2}$ is attainable. so the maximal value of $N_8 = 56$.