Аннотация:On the base of measurements of the rotation periods of 34030 stars from observations with the Kepler space telescope [1] objects with the fastest rotation were studied and estimates of the parameter of their spottedness (considered as indicator of the activity of star) were presented. The properties of two samples of objects containing 670 and 235 stars with P < 1 days and P < 0.5 days, respectively, were obtained and investigated.