Frénicle standard form

A magic square is in Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions apply:

1. the element at position [1,1] (top left corner) is the smallest of the four corner elements; and
2. the element at position [1,2] (top edge, second from left) is smaller than the element in [2,1].

Frénicle's published book of 1693 described all the 880 essentially different order-4 magic squares.

This standard form was devised since a magic square remains "essentially similar" if it is rotated or transposed, or flipped so that the order of rows is reversed — there exists 8 different magic squares sharing one standard form. For example, the following magic squares are all essentially similar, with only the final square being in Frénicle standard form:

 8 1 6   8 3 4     4 9 2   4 3 8     6 7 2   6 1 8     2 9 4   2 7 6
 3 5 7   1 5 9     3 5 7   9 5 1     1 5 9   7 5 3     7 5 3   9 5 1
 4 9 2   6 7 2     8 1 6   2 7 6     8 3 4   2 9 4     6 1 8   4 3 8

Generalising the concept of essentially different squares

384

For each group of magic squares one might identify the corresponding group of automorphisms, the group of transformations preserving the special properties of this group of magic squares. This way one can identify the number of different magic square classes.

From the perspective of Galois theory the most-perfect magic squares are not distinguishable. This means that the number of elements in the associated Galois group is 1. Please compare  A051235 Number of essentially different most-perfect pandiagonal magic squares of order 4n. with  A000012 The simplest sequence of positive numbers: the all 1's sequence.