The navigational behavior of many animals relies on internal representations of space. These representations are often encoded in a world-centric (allocentric) reference frame. To use these representations to guide behavior, animals must transform them to a body-centric (egocentric) reference frame. We took advantage of the experimental tractability of Drosophila melanogaster to determine how neurons in its brain perform this computation. In the fly brain, directions are encoded as vectors in an allocentric reference frame by neurons in a region called the central complex. We investigated whether PFL1, a central complex cell type, transforms allocentric vectors to an egocentric reference frame. Each PFL1 neuron has two dendrites. One of the dendrites receives input from neurons that encode a head direction vector. The other dendrite receives input from neurons in a neuropil called the fan-shaped body (FB). We developed a computational model showing how PFL1 could transform a vector encoded by FB neurons with an allocentric direction, we call the reference direction, to an egocentric reference frame by nonlinearly combining it with the head direction input. To test this model, we simultaneously recorded the head direction input and the output of PFL1 using two-photon microscopy. We used our data to constrain and fit our model, inferring a reference direction. The reference direction is persistent across trials for the same fly but appears random across flies. Our results describe how a population of neurons performs a coordinate transformation, a fundamental computation for any neural system containing representations in multiple reference frames.