$A$ $9\times 9$ real matrix such that $A^3=0$ and $A^{i}\neq 0$ for $i=1$ and $2$. what is the number of possible J.C. Forms(upto similarity) of $A$.

Question

Let $A$ be a $9\times 9$ real matrix such that $A^3=0$ and $A^{i}\neq 0$ for $i=1$ and $2$. Then I want to know what is the number of possible J.C. Forms(upto similarity) of $A$.

We can see that minimal polynomial of $A$ is $x^3$.

So there is a block of order $3$.

Also this the block of maximum order.

so possible JC forms of A are

$J_3, J_3, J_3$

$J_3,J_3,J_2,J_1$

$J_3,J_3,J_1,J_1,J_1$

$J_3,J_2,J_2,J_2$

$J_3,J_2,J_2,J_1,J_1$

$J_3,J_2,J_1,J_1,J_1,J_1$

$J_3,J_1,J_1,J_1,J_1,J_1,J_1$

Here $J_n$ denote the n-block corresponding to eigenvalue 0.

So there are total of 7 possible JC form.

Am I right?