Use formulas for gravitational potential and field at distance $r$ from Earth's center

For a point outside Earth: $V=-\frac{GM}{r}, \qquad g=\frac{GM}{r^2}$

Relate $V$ and $g$

Divide magnitude of potential by field: $\frac{|V|}{g}=\frac{GM/r}{GM/r^2}=r$

So, $r=\frac{|V|}{g}=\frac{5.12\times 10^7}{6.4}$

Calculate $r$

$r=0.8\times 10^7=8\times 10^6\,\text{m}$

Hence, $r=8000\,\text{km}$

Find height above Earth's surface

Earth's radius is $R=6400\,\text{km}$

Therefore height: $h=r-R=8000-6400=1600\,\text{km}$

Match with options

$\boxed{1600\,\text{km}}$

So the correct option is A.