Let's evaluate the given expression step by step. The expression is:

$\left( \frac{1}{2} - \frac{2 \, 1}{4} \right) \div -\frac{2^6}{7} \times 6$

1. First, handle the fraction $\frac{2 \, 1}{4}$, which is a mixed fraction. This is equivalent to $2 + \frac{1}{4} = \frac{9}{4}$. Now, the expression becomes:

$\left( \frac{1}{2} - \frac{9}{4} \right) \div -\frac{2^6}{7} \times 6$

2. Next, simplify the subtraction of the fractions $\frac{1}{2} - \frac{9}{4}$. To do this, we need to express $\frac{1}{2}$ with the same denominator as $\frac{9}{4}$, so $\frac{1}{2} = \frac{2}{4}$. Now subtract:

$\frac{2}{4} - \frac{9}{4} = -\frac{7}{4}$

3. The expression becomes:

$-\frac{7}{4} \div -\frac{2^6}{7} \times 6$

4. Evaluate $2^6$, which equals $64$. Now the expression becomes:

$-\frac{7}{4} \div -\frac{64}{7} \times 6$

5. Division of fractions involves multiplying by the reciprocal of the divisor:

$-\frac{7}{4} \times \frac{7}{64} \times 6$

6. Multiply the fractions:

$\frac{-7 \times 7}{4 \times 64} = \frac{-49}{256}$

7. Now, multiply by 6:

$\frac{-49}{256} \times 6 = \frac{-294}{256}$

8. Simplify $\frac{-294}{256}$ by dividing both the numerator and denominator by 2:

$\frac{-294 \div 2}{256 \div 2} = \frac{-147}{128}$

So, the value of the expression is:

$\boxed{-\frac{147}{128}}$

If you need any further clarification or details, feel free to ask! Here are some related questions for you to explore further:

1. How do you simplify complex fractions like this one?
2. What is the process of multiplying and dividing fractions?
3. How do you handle mixed fractions when performing operations?
4. How do you convert a negative fraction result to its decimal form?
5. What are some rules for working with exponents in expressions?

Tip: When dividing fractions, always multiply by the reciprocal of the divisor.