short answer: yes, not quite.
long answer: redbrick unis aren't especially special anymore with russell group and whatever other groups are now in existence. However, it's hard to place Birkbeck as it's not usually rated in relation to other undergraduate unis but when it is it usually fares quite well and would probably be placed in the top 25 of all universities in the country and possibly fighting for top 10 for certain subjects. Again this is all very contentious as it's department by department and most would say that research reputation is not enough to rate a university on alone. It's in the top 150 in the world I think which is better than a few of the red bricks. Google it to get more.
I'm a current Birkbeck student (albeit a postgraduate student). I don't think you should focus too much on the overall 'reputation' of a University. Give it some weight because reality is that the school does send a signal to employers. However, the strength of the department is FAR MORE IMPORTANT in what you get out of a university in form of competencies. For a number of programmes, Birkbeck is really strong (even top 10), for some others, it really isn't.
I would agree with both comments posted above. I have applied to Brikbeck for an MRes degree course in Structural Biology, which they offer as part of the ISMB in collaboration with UCL. So, that for me is what pulled me to apply there, as their reputation is very strong in this area.
I have also applied to Imperial, King's, UCL and Cambridge, so you see, I rate them very high compared to my other applications.
It depends on the subject and also the ranking used for the subject. It does well on citations too as well as research funding. For example, As far as I understand, Birkbeck wins more funding (approx 60+%) than any other university in the country for its research applications that includes the famous name universities. Also, I believe for certain subjects AND for specifically postgrad, certain areas such as Philosophy are ranked very high - I believe in the top 3 UK institutions for postgrad Philosophy. So it's very difficult to rank universities based on a simple ranking system although sometimes it helps. Other occasions the rankings of some do not help at all. I have seen on the guardian rankings institutions which don't even have a department for that subject ranked as really good. Bananas!