Finding Expressively Adequate truth Functions

You are absolutely right : the exact number is 56, and it can be shown rigorously as follows.

By Post’s theorem, a boolean function is not expressively adequate iff it is either monotone, affine over $\frac{\mathbb Z}{2\mathbb Z}$, self-dual, truth-preserving or falsity-preserving.

In three variables this simplifies considerably. Indeed, not being truth- or falsity-preserving reduces to $f(0,0,0)=1$ and $f(1,1,1)=0$. Then $f$ is never monotone.

We now partition the remaining set of functions into four subsets indexed by $\lbrace 0,1 \rbrace^2$ : for $a,b$ in $\lbrace 0,1 \rbrace$, put

$$ G_{a,b}=\lbrace f | f(0,0,0)=1, f(1,1,1)=0, f(0,0,1)=a, f(0,1,0)=b\rbrace $$

Each $G_{a,b}$ contains a unique affine map (explicitly, $f(x,y,z)=1+ax+by-(a+b)z$ modulo 2). This affine map is always self-dual, and $G_{a,b}$ contains exactly two self-dual maps. Finally, each $G_{a,b}$ contains two nonadequate functions, and hence $2^4-2=14$ adequate functions, which makes a total of $2^2\times (2^4-2)=56$.