Breakdown Voltage and Permittivity

No across the range of materials, but if it is a single material, relationships are there , but to be experimentally found (no formula) as far as I know.

The dielectric constant, also called relative permittivity, is the ratio of flux density to the electric field intensity in a gas, solid, or liquid dielectric.

For a vacuum, the value of ε₀ = 8.85 x 10^-12 farads / meter. Thus, the ratio of the permittivity of any dielectric to that of a vacuum is called the relative permittivity, or ε_r = ε/ε₀. The value of ε for any material then is ε = ε_rε₀. The relative permittivities of various dielectrics are published in many texts, handbooks, etc.

Having said all that, and as pointed out in an earlier post, dielectric constant is only one important aspect, the other is dielectric strength, and that is a function of the specific dielectric, shape of the electrodes, distance between the electrodes, electrical field, voltage waveshape, and environment (temperature and pressure).