I have $P_1 = 24$, $Q_1 = 800000$, $P_2 = 32$ and price elasticity $e = -8$ and need to find the function $P(Q)$.
I guess I need to find $Q_2$ by
$$ \frac{(Q_2-Q_1)/Q_1}{-(P_2-P_1)/P_1} = -8 \Leftrightarrow Q_2 = \frac{8800000}{3} $$
But does it make sense that the quantity will be so much higher by a price increment and thus not follow the law of demand? I know that $e = -8$ is a high price elasticity but this is quite extreme I think.
The good is not a giffen good. These goods have a positive elasticity of demand. Yours has a negative one, by assumption. It cannot yield a giffen good at all.
The error is in the negative sign that you (and also Thomas's answer) assumed.
The formula for the elasticity is:
$$ e_d = \frac{\Delta Q}{\Delta P}\frac{P}{Q}$$
The negativeness of the elasticity is given by the sign of the derivative (a price increase leads to a fall in quantity demanded; the contrary in a Giffen good).
In your example, assuming as starting point $Q_1$ and $P_1$ (you could alternative use the end point, or the arc elasticity; see here), you get:
$$ -8 = \frac{Q_2 - 800,000}{12}\frac{24}{800,000}$$
Which yields
$$Q_2 = -2,400,000$$
This is, in effect, a negative quantity. To see why this is the case, rewrite the elasticity function as follows:
$$ e_d \frac{\Delta P}{P_1} = \frac{\Delta Q}{Q_1}$$
This is, the percentage change in the quantity is equivalent to the percentage change in prices times the elasticity.
Replacing the numbers, you get:
$$ -4 = \frac{\Delta Q}{Q_1}$$
This is, the quantity falls by 400% percent!. That is why it becomes negative (from 800,000 to -2,400,000).
The origin of the problem is the massive change in price (50%) together with the massive elasticity of demand (-8), which gives the total change in quantity ($50\% \times -8 = -400\%$). A less dramatic combination would yield a positive $Q_2$.
You have that price elasticity is defined by:
$$\varepsilon_{d} = -\frac{\partial Q}{\partial P}\frac{P}{Q} \approx -\frac{P}{Q}\frac{\Delta Q}{\Delta P}$$
Initially we have $P = P_{1}$ and $Q=Q_{1}$ and after the perturbation we have $\Delta P = (P_{2} - P_{1})$ and $\Delta Q = (Q_{2} - Q_{1})$, therefore:
$$-\frac{P_{1}}{Q_{1}}\frac{Q_{2}-Q_{1}}{P_{2}-P_{1}}=-8 \implies Q_{2}-Q_{1}=\frac{8Q_{1}(P_{2}-P_{1})}{P_{1}}$$
And therefore:
$$ Q_{2}=Q_{1}+\frac{8Q_{1}(P_{2}-P_{1})}{P_{1}}=\frac{8800000}{3} $$
This is so large, partly because the price elasticity is so large, but also probably because of the error in the finite difference approximation method over such large changes in $P$.
