The average score of the group of students is given as 3.5. To find the value of \(x\), we can use the formula for the average:

\[

\text{Average} = \frac{\text{Sum of all scores}}{\text{Total number of students}}

\]

Given that the scores are 1, 2, 3, 4, 5, and 6, and the corresponding number of students are 1, 4, 5, 6, \(x\), and 2, we can write:

\[

3.5 = \frac{1 \cdot 1 + 2 \cdot 4 + 3 \cdot 5 + 4 \cdot 6 + 5 \cdot x + 6 \cdot 2}{1 + 4 + 5 + 6 + x + 2}

\]

Now, solve for \(x\):

\[

3.5 = \frac{1 + 8 + 15 + 24 + 5x + 12}{18 + x}

\]

Multiply both sides by \(18 + x\):

\[

3.5 \cdot (18 + x) = 1 + 8 + 15 + 24 + 5x + 12

\]

Distribute:

\[

63 + 3.5x = 60 + 5x

\]

Subtract \(3.5x\) from both sides:

\[

63 = 60 + 1.5x

\]

Subtract 60 from both sides:

\[

3 = 1.5x

\]

Finally, solve for \(x\):

\[

x = \frac{3}{1.5} = 2

\]

So, the value of \(x\) is 2.