What is the formula for the gain factor of a ferromagnetic core for a torquer coil rod?

I am an undergraduate who is currently designing an actuation system for a CubeSat, more specifically a PCB magnetorquer. (with 2 ferromagnetic core coils and one air core)

I made an excel sheet that helps me determine the magnetic dipole moment and more from some sets of data including the size and material the core is made of. The material of choice is permalloy.

I supposed it is known that the magnetic dipole moment is a function of the intensity of the current, the area of the core , the number of turns AND the gain factor $K$:

${mu} = k N I A$

where $N$ is the number of turns, $I$ is the current value and $A$ vector cross-sectional area of the solenoid.

Unfortunately, I found little information online about this gain factor which is taken as $K=1$ for an air core coil, but it varies between values of 100 and 300 for a ferromagnetic core, depending on the length and diameter shape factor and permeability of the material.

That is all that I’ve found online about this gain factor and it is the reason why I am asking you guys to provide a general formula of this factor or at least its value for permalloy.

Another thing that is worth mentioning is that I had a conversation last year about this value with an academic researcher and confirmed me the existence of this it, but we did not developed further on the subject.

To be noted, there is a typo problem in the pic attached, it should be $k=100$ to $300$.

The thesis where i found this "K" value, i can attach links to more thesis where this gain factor is present if needed