The substitution and permutation function, which are presented in the article as elements of a number of factorial sets, are the key functions of cryptographic systems that provide diffusion and mixing of information. A new scale of notation is proposed while analyzing this problem. This is the notation scale of a number of factorial sets. This scale of notation helps to index the elements of a number of factorial sets and establish a one-to-one correspondence between the number and a specific type of substitution. This allows analyzing substitutions characteristics systematically. This paper presents the basic concepts of a number of the factorial sets. It is noted that the permutations of the factorial sets form symmetric permutation groups, and specific permutations (when raised to a power) form cyclic groups. The group axioms are fulfilled for the permutations of a number of factorial sets. Also, the definition domain, the group operation of multiplication, and identical and inverse substitutions are given for them. The number of independent cycles, decrement, inverse, parity and sign are common characteristics of the substitutions of a number of factorial sets. The criteria for choosing single substitutions with the best characteristics are proposed.