Air Carrier ERJ170 Captain reported encountering wake turbulence in trail of a B777 climbing through FL275. Reporter stated there was no alert from ATC prior to the encounter.

Narrative

Climbing out of DCA had multiple speed/HDG (Heading) changes for spacing. Climbing up to FL270 had speed assigned of 270 knots. Switched to freq of XXX.XX and was told to maintain 290 knots. There had been weather all day and rides weren't the best so ATC has a canned answer of no bad rides which unfortunately is normal now. Climbing out we were IMC and it was not smooth but light chop. As we were climbing ATC asked us if we could climb to FL290 and we could so were assigned FL290. As we were climbing through FL270 the ride started to deteriorate but I blamed it on getting in the tops of the clouds and I noticed we were climbing at the same rate as the plane 5 miles ahead of us on TCAS and didn't think much of it. At about FL275 we get above the clouds and I could physically see the plane in front of us and I knew something didn't look correct and about the same time we hit their wake and it was bad - it was a Boeing 777. We tossed from left to right pretty violently and I tried to get the plane under control never got a bank angle warning. I slowed down and immediately slowed the climb rate to stay below their path. I asked for off course heading and was given 10 left or right. We were directly downwind of the 777 wake turbulence but still had visual so I went left. I asked why were not given a heads up of this plane and to which I didn't get an answer. I asked again and there was a controller change. Did someone mess up? I had my FO (First Officer) call to the back to see if everyone was OK and Flight Attendants said they were okay. Once on ground I was informed by the A Flight Attendant that she was tossed to the ground and fell on her hands and knees but she was OK. Hitting wake turbulence at FL280 could have resulted in a much worse result and thankfully it didn't.