You've seen the nut, no doubt?

One surface is pretty close to 27mm. The other is about 32mm. Those are the official sizes, but neither is a perfect fit. The sides of the nut are rounded. Te flat areas are not very large.

I ended up buying one of the PSR sockets that will grab the 27 and 32 surface a the same time. The socket is expensive, but it works. Replacement nuts are not cheap. The first one I removed had enough damage to where I didn't want to use it again.

I do have an old 1.25" impact socket that fit fairly well. Il post a pic of the oar socket. A few years ago the price was reasonable, but now it's 60.00+

 

Here that would be kidney selling price....
By 27 and 32... Do you mean one of those tower shaped nuts...like a wedding cake?
Pretty certain mine is just a plainish nut just by looking up along the chain to the sprocket