which or what kind of star has a very stable luminosity? I mean those kind of stars should be very stable and barely have activity.

Their light curves should be very very flat.

Our sun, as a G type star, are not so stable. At lest there are lots of floating solar spots.
Hipparchos numbers of the 26 most stable stars known:
2021, 2854, 5542, 16611, 19747, 24927, 32537, 38414, 42913, 45556, 50191, 57363, 71053, 73555, 74666, 76440, 74946, 90139, 94648, 96052, 10239, 102488, 104732, 111169, 116631, 118322.

Compare Hpmax and Hpmin: They differ 0.01. Standard error e_Hpmag is 0.0003 mag.
There are 681 stars in the Hipparchos catalog with an amplitude of 0.01 mag.

More detail in this paper, including a link to the appropriate VizieR query.