  This article shows the asymptotics of distributions of various functionals of
the Beta$(2-\alpha,\alpha)$ $n$-coalescent process with $1<\alpha<2$ when $n$
goes to infinity. This process is a Markov process taking {values} in the set
of partitions of $\{1, \dots, n\}$, evolving from the intial value
$\{1\},\cdots, \{n\}$ by merging (coalescing) blocks together into one and
finally reaching the absorbing state $\{1, \dots, n\}$. The minimal clade of
$1$ is the block which contains $1$ at the time of coalescence of the singleton
$\{1\}$. The limit size of the minimal clade of $1$ is provided. To this, we
express it as a function of the coalescence time of $\{1\}$ and sizes of blocks
at that time. Another quantity concerning the size of the largest block (at
deterministic small time and at the coalescence time of $\{1\}$) is also
studied.
