G6 civil servant here - i assume you are not currently in the CS? Sorry to hear you were unsuccessful. It is very common to not be successful on your first application to the CS due to the way recruitment is done. Apologies if some of the below seems basic but they are common errors i have seen in 8 years of being interviewed and interviewing (i joined as HEO and worked my way up to G6). If you wanted to give a bit more detail about the Dept or role, then maybe i could give some more detailed help? (Although appreciate you might not want to if too outing).
As @emsyj37 says your application needs to be written in reference to the success profiles. The job listing will list which of these are being assessed. You should not just look at the G7/6 level but also look at SCS1 level and incorporate elements of those if you can.
You will need to demonstrate that you meet each of the criteria listed in the person specification - in the sift, points are awarded for meeting each criteria. Even if it's a just a line eg. "i am proficient in microsoft office / i am an advanced excel user etc etc" in relation to a requirement to the person spec - if there has been a lot of applicants, it could be the difference between being sifted or not.
If you were asked to provide examples of certain behaviours, did you use the STAR technique? I see a lot of applications where candidates (for want of a better term) spend a lot of time on the S and T, but not a lot of time on the A and R, which is where you want to focus your answer.
Fundamentally though, in the last recruitment i ran before i went on mat leave, we received over 100 applicants for an SEO role position, of which we interviewed 8 and recruited 2. The numbers tend to be lower at the more senior grades, but you also don't get the multi-vacancy positions so there is just as much competition for the posts. So it can just be a question of numbers and where 1 or 2 points in the sift can make all the difference.