Coin placement game on a symmetric table

Two players alternately place identical quarters flat on a table, with no coin allowed to overlap another. The table can be any size, but it is symmetric about its centre (like a round table). The first player who has no legal spot for a coin loses. Would you rather move first? Answer $1$ for yes, $2$ for no.

1 (go first). Put your first coin exactly at the centre. After that, whenever your opponent places a coin, place yours at the point reflected through the centre; that spot is always free, so you always have a reply and your opponent runs out of moves first.