To find the number of conjugate classes in a non-abelian group of order 729, we need to understand the concept of conjugacy in group theory.



In group theory, two elements a and b of a group G are said to be conjugate if there exists an element g in G such that b = gag⁻¹. In other words, two elements are conjugate if they are related by a change of basis.



A conjugate class is a set of elements in a group that are all conjugate to each other. Each element of a conjugate class is called a representative of the class.



In a non-abelian group, the number of conjugate classes is equal to the number of distinct orbits of the conjugation action of the group on itself.

Let's denote the number of conjugate classes in a group G as n(G).

To find n(G), we can use the class equation:

|G| = |Z(G)| + Σ |G : C(g)|,

where |G| represents the order of the group G, |Z(G)| represents the order of the center of G, Σ represents the sum over all distinct conjugate classes, and |G : C(g)| represents the index of the centralizer of an element g in G.

In our case, |G| = 729, and since G is non-abelian, |Z(G)| ≠ |G|.



The order of the group G is 729, which can be written as 3^6. In general, the order of a group can be expressed as a product of prime powers.



We need to find the number of distinct conjugate classes, which is equal to the number of terms in the sum on the right-hand side of the class equation.

Since |G : C(g)| is a positive integer for each conjugate class, the sum must be a multiple of 3.

The prime factorization of 729 shows that the order of the group G is divisible by 3, which means that the sum in the class equation must also be divisible by 3.

However, |Z(G)| ≠ |G|, so |Z(G)| cannot contribute a multiple of 3 to the sum. Therefore, the number of terms in the sum must be a multiple of 3.

Since the number of terms in the sum represents the number of distinct conjugate classes, we can conclude that the number of conjugate classes in a non-abelian group of order 729 is a multiple of 3. However, without further information about the specific group G, we cannot determine the exact number of conjugate classes.