Very symmetric convex polytope Let $C_n$ be the convex polytope in ${\mathbb R}^n$ defined by the inequalities
(in $n$ variables $x_1,x_2, \ldots ,x_n$) :
$$
x_i \geq 0, x_i+x_j \leq 1
$$
(for any indices $i<j$). 
Denote by $E_n$ the set of extremal points of $C_n$. We have a natural action
of the symmetric group $G={\mathfrak S}_n$ on $C_n$, and hence on $E_n$ also. So we have
a quotient set $\frac{E_n}{G}$. Here are some questions about it, in decreasing order
of difficulty :
1) Is a simple description of $\frac{E_n}{G}$ known in general ?
2) What is the asymptotic behaviour of the sequence $(|\frac{E_n}{G}|)_{n \geq 2}$ ?
3) Is the sequence $(|\frac{E_n}{G}|)_{n \geq 2}$ bounded ?

Perhaps I'm overloooking something, but it seems to me that the extreme points of $C_n$ are of three sorts.  (1) The zero vector.  (2) The standard unit vectors, with a $1$ in a single component and zeros in the other $n-1$ components.  (3) Vectors with the entry $1/2$ in some set of three or more components and zeros in the remaining components.  If that's right, then there are a single $G$-orbit for (1), another for (2), and $n-2$ orbits for (3).  Each orbit for (3) is characterized by the cardinality, between $3$ and $n$ inclusive, of the set of coordinates where $1/2$ occurs.
