Number of similarity classes of matrices $A$ in $M_6(\mathbb{R})$ satisfying $(A-2I)^3=0.$

Question

Find the number of similarity classes of matrices $A$ in $M_6(\mathbb{R})$ satisfying $(A-2I)^3=0.$

If $(A-2I)^3$ then the characteristic polynomial is $p_A(x)=(x-2)^6$ and we have 3 posibilities for $m_A(x)$, wich are: $(x-2)^3, (x-2)^2, (x-2)$.

We have then:

1. if $p_A(x)=(x-2)^6$ and $m_A=(x-2)^3$ then

$A=\begin{bmatrix} 2 & & & & & \\ 1 & 2 & & & & \\ & 1 & 2 & & & \\ & & & 2 & & \\ & & & 1 & 2 & \\ & & & & 1 & 2 \\ \end{bmatrix}$ $A=\begin{bmatrix} 2 & & & & & \\ 1 & 2 & & & & \\ & 1 & 2 & & & \\ & & & 2 & & \\ & & & 1 & 2 & \\ & & & & & 2 \\ \end{bmatrix}$ $A=\begin{bmatrix} 2 & & & & & \\ 1 & 2 & & & & \\ & 1 & 2 & & & \\ & & & 2 & & \\ & & & & 2 & \\ & & & & & 2 \\ \end{bmatrix}$

2. if $p_A(x)=(x-2)^6$ and $m_A=(x-2)^2$ then

$A=\begin{bmatrix} 2 & & & & & \\ 1 & 2 & & & & \\ & & 2 & & & \\ & & 1 & 2 & & \\ & & & & 2 & \\ & & & & 1 & 2 \\ \end{bmatrix}$ $A=\begin{bmatrix} 2 & & & & & \\ 1 & 2 & & & & \\ & & 2 & & & \\ & & 1 & 2 & & \\ & & & & 2 & \\ & & & & & 2 \\ \end{bmatrix}$ $A=\begin{bmatrix} 2 & & & & & \\ 1 & 2 & & & & \\ & & 2 & & & \\ & & & 2 & & \\ & & & & 2 & \\ & & & & & 2 \\ \end{bmatrix}$

3. if $p_A(x)=(x-2)^6$ and $m_A=(x-2)$ then

\begin{bmatrix} 2 & & & & & \\ & 2 & & & & \\ & & 2 & & & \\ & & & 2 & & \\ & & & & 2 & \\ & & & & & 2 \\ \end{bmatrix}

Is this correct?