Boolean functions with a vertex-transitive group of automorphisms

A Boolean function is called vertex-transitive, if the partition of the Boolean cube into the preimage of 0 and the preimage of 1 is invariant under a vertex-transitive group of isometric transformations of the Boolean cube. The logarithm of the number of the vertex-transitive functions of n variables is at least Ω(n) and at most O(n logn). There is a polynomial over F2 of any given degree, which defines a vertex-transitive function, and quadratic polynomials with this property can be characterized. There is a vertex-transitive function of n = 4 variables with sensitivity n. Some properties of the groups of the automorphisms of the vertex-transitive functions are presented.