There are long-standing achievement gaps in England associated with socio-economic status (SES), ethnicity and gender, but relatively little research has evaluated interactions between these variables or explored school effects on such gaps. This paper analyses the national test results at age 7 and age 11 of 2,836 pupils attending 68 mainstream primary schools in an ethnically diverse inner London borough. The groups with the lowest educational achievement and poorest progress were both Black Caribbean "and" White British low SES pupils. White British middle and high SES pupils made substantially more progress than White British low SES pupils, significantly increasing the SES gap over time. However low and high SES Black pupils made equally poor progress age 7-11. School effects on pupil progress were large, but there was no evidence of differential school effectiveness in relation to SES, ethnicity or gender. Low SES pupils in the more effective schools performed significantly better than high SES pupils in the less effective schools, but all pupils (both low and high SES) benefit from attending the more effective schools and so these schools do not eliminate the SES gap. The limits to change that may be achieved by schools alone are discussed.