How is rearranging $56\times 100\div 8$ into $56\div 8\times 100$ allowed by the commutative property? So according to the commutative property for multiplication:

$a \times b = b \times a$ 

However this does not hold for division

$a \div b \neq b \div a$ 

Why is it that in the following case:

$56 \times 100 \div 8 = 56 \div 8\times 100$

It seems like division is breaking the rule. There is something I am misunderstanding here. 

Is it because $a\times b\div c=a\div c\times b$ is allowed since $b\div c$ are not being rearranged so that $c\div b$? 

If this is the case are you allowed to rearrange values in equations so long as no values have the form $a \div b = b \div a$ and $a - b = b -a$ ?
Notice that you have always $\div 8$, no matter the order of the other terms. You don't divide by a different number. It might help to think $\div c$ as a multiplication with $d=1/c$. Then everything would look easier:
$$a\times b\div c=a\times b\times d=a\times d\times b=a\div c\times b$$
With multiplication and division alone (no addition or subtraction) they are associative with respect to one another, BUT division itself is NOT commutative.

So basically you can do the multiplication or division in either order, BUT you must respect which way you interpret the inputs to the left or right of the operator's sign. If you flip this it has the effect of flipping the inputs which is equal precisely when the operator is commutative. As you noted, with division, it is not commutative.