We are currently constructing a prototype quadrupole electromagnet for the proposed Induction Linac Systems Experiment (ILSE) at LBNL. ILSE will address many physics and engineering issues relevant to the design of a heavy-ion fusion driver accelerator. The pulsed electromagnet has two layers of current windings and will produce a field gradient of 28 T/m, with a usable aperture of 6 em. It operates at a repetition rate of 1 Hz, steady-state. In this paper, we discuss how the interaction of various concerns such as maximum dynamic ap~rture, short lattice period, field quality, iron yoke weight, heat transfer, and voltage standoff have led to our particular design choices. We also present 2-and 3-D numerical calculations concerning field topography and the results of transport simulations of space-charge dominated ion beams with ILSE parameters.