Introduction:
The group of all symmetries of a square, denoted as G, is a finite group. In this group, each element represents a symmetry of the square, such as rotations or reflections. Conjugate classes in a group are sets of elements that are related to each other through conjugation by other elements in the group. In this case, we need to determine the number of conjugate classes in G.

Identifying the symmetries:
Before we proceed to determine the number of conjugate classes, let's first identify the different symmetries of a square. The symmetries of a square include:

1. Identity symmetry (e): This symmetry leaves the square unchanged.
2. Rotations (r1, r2, r3, r4): There are four possible rotations - 90 degrees clockwise, 180 degrees, 270 degrees clockwise, and 360 degrees (identity).
3. Horizontal reflections (h1, h2): There are two possible horizontal reflections.
4. Vertical reflections (v1, v2): There are two possible vertical reflections.
5. Diagonal reflections (d1, d2): There are two possible diagonal reflections.

Determining conjugate classes:
To determine the conjugate classes in G, we need to consider the conjugacy relation. Two elements 'a' and 'b' in a group G are said to be conjugate if there exists an element 'g' in G such that g⁻¹ag = b.

Let's consider each symmetry and determine its conjugate class:

1. Identity symmetry (e): Since the identity symmetry is unchanged under conjugation, it forms its own conjugate class.

2. Rotations (r1, r2, r3, r4): Any rotation can be obtained by conjugating any other rotation by the appropriate power of another rotation. Therefore, each rotation forms its own conjugate class.

3. Horizontal reflections (h1, h2): One horizontal reflection can be obtained by conjugating the other horizontal reflection by a rotation. Therefore, both horizontal reflections belong to the same conjugate class.

4. Vertical reflections (v1, v2): Similar to horizontal reflections, one vertical reflection can be obtained by conjugating the other vertical reflection by a rotation. Therefore, both vertical reflections belong to the same conjugate class.

5. Diagonal reflections (d1, d2): One diagonal reflection can be obtained by conjugating the other diagonal reflection by a rotation. Therefore, both diagonal reflections belong to the same conjugate class.

Counting the conjugate classes:
From the analysis above, we have the following conjugate classes:

1. Identity symmetry (e)
2. Rotations (r1, r2, r3, r4)
3. Horizontal reflections (h1, h2)
4. Vertical reflections (v1, v2)
5. Diagonal reflections (d1, d2)

Therefore, there are a total of 5 conjugate classes in the group of all symmetries of a square. Hence, the correct answer is option 'C' (6).