The brain must recognize objects in the face of identity-preserving transformations such as changes in lighting, position, and orientation. However, to achieve general purpose geometric reasoning these view transformations should also be represented in neural codes. A natural strategy to encode both the identity of an object and its view is to utilize an equivariant code, where the neural representation transforms in a manner consistent with the transformations to the inputs, and consistent across objects. A classic example of such a code is given by an intermediate layer of a convolutional network, which is equivariant to spatial translations of the inputs. A fundamental and unanswered question is how equivariant structure in a code alters the number of objects that can be expressed (capacity). To address this, we derive a complete theory of perceptron capacity, which measures the number of objects that can be linearly separated under all possible labelings, for equivariant neural codes and apply this theory to models of vision. We show that our theory accurately predicts the capacity of simple models of visual cortex and convolutional networks, showing that capacity scales not with the number of neurons in the circuit but rather with the number of trivial irreps of the representation. These results constitute an advance in the theory of the expressivity of learning systems under the natural condition of equivariance.