Consider the collection of symmetric groups $\{\Sigma_n\}_{n\geq1}$ as a semi-simplicial set (i.e. a simplicial set without degeneracies) as follows. Consider $i\in\{1,\dots,n+1\}$ and $\pi\in\Sigma_{n+1}$ represented as a sequence $(\pi(1),\dots,\pi(n+1))$, then $$d_{i-1}(\pi)=(\pi(1)-\epsilon_1,\dots, \widehat{\pi(i)},\dots,\pi(n+1)-\epsilon_{n+1})$$ where for every $j\in\{1,\dots,n+1\}$ $$\epsilon_j=\begin{cases} 0 & if\ \ \pi(j)<\pi(i)\\ 1 & if\ \ \pi(j)>\pi(i). \end{cases}$$

I am wondering what it's known about the homotopy type of the geometric realization of this semi-simplicial set?