Been through this. Your washer will not for long pump water to the first floor. The washer pumps are not designed to do this. The pumps are designed to run no more then 3.5' above the top of the appliance. Plus if you do this the old grey water from the previous wash that is left in the pipe can leak back into the washer. Even if you put a check valve in.
You can run the water to the sump crock if you have one. The sump pump will remove the water to where it is going when it rains. Sump pumps will handle washer water well. This may not be code in some areas where rain sump water is not allowed in sanitary sewers or where it is draining into a storm sewer. It's been done many times.
Another idea if your sewer pipes run in the basement under the first floor. Do a measurement. Many times the top of the washer will be less then 3.5 feet (42 inches) up from the lowest sewer pipe. In that case you can tap into the pipe and install an inline check valve. You can also make a platform to stand on with the washer raised to get the washer up high enough.
Still another idea is to buy a simple utility sink or a sump pump crock and bricks to put the sump pump crock on at the home improvement store (cost is $20). Plug the bottem of sink (if you use one) and set a sump pump in the bottem of the sink or crock. Connect the piping to the first floor sewers and the sump pump will do the work. I have had to do this in rental properties for cheap landlords. Works great. Total cost about $150 in parts. This is code as well. Ugly but legal.
In closing all of these suggestions are assuming your sewer pipes aren't running in the basement through the basement floor.