Supermassive black holes with masses of millions to billions of solar masses are commonly found in the centers of galaxies. Astronomers seek to image jet formation using radio interferometry but still suffer from insufficient angular resolution. An alternative method to resolve small structures is to measure the time variability of their emission. Here we report on gamma-ray observations of the radio galaxy IC 310 obtained with the MAGIC (Major Atmospheric Gamma-ray Imaging Cherenkov) telescopes, revealing variability with doubling time scales faster than 4.8 min. Causality constrains the size of the emission region to be smaller than 20% of the gravitational radius of its central black hole.We suggest that the emission is associated with pulsar-like particle acceleration by the electric field across a magnetospheric gap at the base of the radio jet.