In this paper, an effective generalized spatial modulation (GenSM) scheme with transmit antenna grouping is proposed to overcome the performance degradation caused by correlated channels. In the proposed scheme, the transmit antennas are divided into several equal-sized groups, and spatial modulation (SM) is carried out to select one active antenna in each group independently. It is quite different from the conventional GenSM which jointly selects active antenna set. Apart from the straightforward block grouping method, which collects the adjacent antennas to the same group, interleaved grouping is also introduced. It can maximize the average distance between the antennas in the same group, since the channel correlation depends on it. To evaluate the performance, a closed-form expression of the average bit error probability (ABEP) upper bound is derived for all proposed grouping methods and Monte-Carlo simulations are conducted to verify the analysis and reveal the performance gain of the proposed scheme in terms of bit error rate (BER) in comparison with conventional GenSM and SM.