All different functions

Question

For functions \$f, g: \{0,1\}^n \rightarrow \{0,1\} \$, we say \$f \sim g\$ if there's a permutation of \$1,2,3,...,n\$ called \$i_1,i_2,i_3,...,i_n\$ so that \$f(x_1,x_2,x_3,...,x_n) = g(x_{i_1},x_{i_2},x_{i_3},...,x_{i_n})\$. Therefore, all such functions are divided in several sets such that, for any two functions \$f, g\$ in a same set, \$f \sim g\$; for any two functions \$f, g\$ in different sets, \$f \not\sim g\$. (Equivalence relation) Given \$n\$, output these sets or one of each set.

Samples:

0 -> {0}, {1}
1 -> {0}, {1}, {a}, {!a}
2 -> {0}, {1}, {a, b}, {!a, !b}, {a & b}, {a | b}, {a & !b, b & !a}, {a | !b, b | !a}, {a ^ b}, {a ^ !b}, {!a & !b}, {!a | !b}

You can output the function as a possible expression(like what's done in the example, but should theoretically support \$n>26\$), a table marking outputs for all possible inputs (truth table), or a set containing inputs that make output \$1\$.

Shortest code win.

Answer 1

J, 62 bytes

f=:3 :'~.(2#~2^y)([:<@~.@/:~((i.@!A.i.)y)|:(y#2)$])@#:i.2^2^y'

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-3 bytes for anonymous function (removing f=:)

For each boolean function (truth table), generate its equivalence class, then remove duplicates.