The ALPHA-g experiment at CERN aims to undertake the first ever
precision measurement of the gravitational mass of antihydrogen atoms,
by releasing them from a gravity-compensated magnetic minimum trap.
The magnetic minimum trap is nominally created by an octupole and two
mirror coils. However, in order to achieve the level of magnetic
control required to resolve gravity to the 1% level and beyond, many
other magnetic elements are involved to tailor and correct the trap
field to the O(10^-5) level, as well as to manipulate and cool the
anti-atoms before the measurement. In this talk, we present the
construction of the sophisticated, multi-purpose superconducting
magnet system designed for this purpose, which composes of 24 coils, 5
overlapping octupole elements and a high-uniformity large volume
solenoid. We will also showcase the simulated behaviour of the
anti-atoms inside the magnet system, as well as the predicted gravity
precision we may achieve under different experimental conditions.