  Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of
characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that
$D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over
$k$ of the same codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts
of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the
numbers $\gamma_D(i):=\dim(\pmb{Aut}(D[p^i]))$. We show that $n_D\le cd$,
$(\gamma_D(i+1)-\gamma_D(i))_{i\in\Bbb N}$ is a decreasing sequence in $\Bbb
N$, for $cd>0$ we have $\gamma_D(1)<\gamma_D(2)<...<\gamma_D(n_D)$, and for
$m\in\{1,...,n_D-1\}$ there exists an infinite set of truncated Barsotti--Tate
groups of level $m+1$ which are pairwise non-isomorphic and lift $D[p^m]$.
Different generalizations to $p$-divisible groups with a smooth integral group
scheme in the crystalline context are also proved.
