A central idea in understanding brains and building artificial intelligence is that structure determines function. Yet, how the brain's complex structure arises from a limited set of genetic instructions remains a key question. The ultra high-dimensional detail of neural connections vastly exceeds the information storage capacity of genes, suggesting a compact, low-dimensional blueprint must guide brain development. Our motivation is to uncover this blueprint. We introduce a generative model, to learn this underlying representation from detailed connectivity maps of mouse cortical microcircuits. Our model successfully captures the essential structural information of these circuits in a compressed latent space. We found that specific, interpretable directions within this space directly relate to understandable network properties. Building on this, we demonstrate a novel method to controllably generate new, synthetic microcircuits with desired structural features by navigating this latent space. This work offers a new way to investigate the design principles of neural circuits and explore how structure gives rise to function, potentially informing the development of more advanced artificial neural networks.