Geometric interpretation of $2A$ conjugacy class in Conway group $Co_1$

Question

I am struggling with following problem. Consider $2A$ class in $Co_1$ having $819*759*75$ elements. Each element $a$ from $2A$ have two representatives in $Co_0$. Element $a$ corresponds to $E_8$ sublattice in Leech lattice defined as $\{v: av=-v\}$ where I call by $a$ also proper preimage in $Co_0$. Now the opposite: having $E_8$ sublattice $L$ in Leech lattice I can find element $a_L$ in $2A$ class.

My goal is to find relation between $Order(ab)$ for $a,b$ in $2A$ and corresponding geometry of two $E_8$ sublattices. The $Order(ab)$ can be $2,3,4,5,6$. (BTW in 2017 I have found this relation already).

Take any other sporadic group $g$ and certain conjugacy class $C$ of involutions. The possible values of $Order(ab)$ for $a,b$ in $C$ can be obtained from character table. Can this help to connect some "lattice" to the group $g$ ?

Answer 1

Here is the answer for $Co_1$ which I obtained in GAP.

For given $2A$ involution $a$ in $Co_1$ the number $n$ of such $2A$ involutions $b$ that product $ab$ is in conjugacy class $C$ is presented in format $[C, n]$. I call hook the order of product.

$[ [ "1a", 1 ], [ "2a", 12870 ], [ "2c", 60480 ], [ "3b", 573440 ], [ "3d", 491520 ], [ "4a", 34560 ], [ "4c", 4838400 ], [ "4d", 2419200 ], [ "5b", 12386304 ], [ "6e", 25804800 ] ]$

Next the relation between two $E_8$ sublattices $L_a$ and $L_b$ of Leech lattice is presented in following format. [hook of a and b, number of vectors belonging to intersection of $L_a$ and $L_b$, dimension of vector space generated by both $L_a$ and $L_b$, $[k,l]$ where $k$ is number of vectors in $L_a$ having exactly $l$ perpendicular vectors in $L_b$.

First two rows correspond to $2A$ hook in $Co_1$. These are two cases in $Co_0$. For given $E_8$ sublattice there are $270$ sublattices perpendicular to it (first row) and $12600$ intersecting in $D_4$ sublattice (second row).

$[ 2, 0, 16, [ [ 240, 240 ] ] ]$

$[ 2, 24, 12, [ [ 192, 84 ], [ 24, 126 ], [ 24, 240 ] ] ]$

$[ 2, 4, 14, [ [ 48, 84 ], [ 132, 126 ], [ 60, 240 ] ] ]$

$[ 3, 0, 16, [ [ 240, 126 ] ] ]$

$[ 3, 6, 14, [ [ 162, 84 ], [ 78, 126 ] ] ]$

$[ 4, 0, 16, [ [ 240, 84 ] ] ]$

$[ 4, 0, 16, [ [ 24, 84 ], [ 192, 126 ], [ 24, 240 ] ] ]$

$[ 4, 2, 15, [ [ 120, 84 ], [ 114, 126 ], [ 6, 240 ] ] ]$

$[ 5, 0, 16, [ [ 120, 84 ], [ 120, 126 ] ] ]$

$[ 6, 0, 16, [ [ 66, 84 ], [ 168, 126 ], [ 6, 240 ] ] ]$

The numbers $84$ and $126$ which appear on second $l$ position correspond to $D_7$ and $E_7$ sublattices of $E_8$. The first case happens when projection of the vector on $E_8$ is collinear with "type 4" vector which is sum of two perpendicular vectors from the lattice. The second case happens when projection of the vector on $E_8$ is collinear with lattice vector.

As we can see one of the 4-hook is obtained by intersection in one line of the two 8-planes. This feature can be used to define the Leech lattice from $Co_0$ in 24-dim space. I don't know whether this is useful for something yet.