Frénicle standard form

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

  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].

In 1693, Frénicle described all the 880 essentially different order-4 magic squares.[1]

Properties

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 exist 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 the 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

Generalizations
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 (enumerated in OEIS: A051235) are not distinguishable since the size of the associated Galois group is 1.

gollark: Point is that it's not exactly a very significant hurdle for whoever is designing weapons, but is quite annoying for people working on cubesats and balloons and stuff.
gollark: It's a weird restriction, considering that presumably if you can engineer an entire missile you can also work out a way around restrictions in GPS hardware, to be honest.
gollark: Apparently the US was worried about GPS being used by enemy ICBMs (???) so now consumer GPS devices will refuse to work above certain speeds/heights.
gollark: You can do GPS with RTL-SDRs apparently, which gets around the weird height/speed restrictions in consumer devices.
gollark: There's interesting stuff with satellites and whatnot, but that needs a lot of hardware.

References
  1. B. Frénicle de Bessy; et al. (1693). Divers ouvrages de mathematique et de physique.
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