Socioeconomic status is related to a number of factors that are complicated so it's difficult to tell what causes what.
Parents with low socioeconomic status are less likely to breastfeed, they're less likely to provide nutritional variety, they're less likely to read to their children, they talk to their children less, and they're more likely to use discipline techniques that contribute to antisocial behavior. It's thought that if such parents were educated about breastfeeding, nutrition, reading, and discipline, that the children will receive the benefits of the type of education that helps a person from the very beginning - infancy, toddlerhood, etc. All of these things have been shown to make people more intelligent (that is, have higher IQ's) and discipline obviously shapes our sense of right and wrong. One theory is that the stress of being low in socioeconomic status lessens the amount of positive parent/child interaction. So it's a cycle.
I think it has less to do with formal education because when you see behavioral problems, it's usually very early on. And when a child is identified as having behavioral problems early on, he/she's more likely to drop out to work a working class job. Obviously there are some who graduate and still go into working class jobs because they like or need them, and these people are certainly no less intelligent (on an IQ level). Going for a higher level education can be tough for someone with low socioeconomic status, and a college degree doesn't necessarily imply intelligence; it implies higher ambition and drive and opportunity.
So it may be that the "unintelligent" working class that you're speaking of (the ones that use their fists) may be the product of a household low in parental education. And, yes, the tension is probably caused by things that the working class specifically has to worry about - layoffs, lack of benefits, low pay. But not all working class jobs are like that. Stress can make people become alienated.
So I'd say that you're right to make the link, but it's not that say "Dumb people are violent." Or "Working class people are dumb and violent." It's more like "The upbringing of people affects their later decisions about education, violence, and work ethic."
I want to point out that I'm in college right now and my boyfriend works in a factory. He's never been violent in his life; he just likes it. He was raised in a loving environment and his mother has run a daycare for most of her life, so she's taken countless classes and her parenting style is a result of that.
I know it was long, but I just wanted people to really think about putting cause and effect in the right place.