We present IRAM 30m telescope observations of the CO ( 1-0 ) and ( 2-1 ) lines in a sample of 11 group-dominant elliptical galaxies selected from the CLoGS nearby groups sample . Our observations confirm the presence of molecular gas in 4 of the 11 galaxies at > 4 \sigma significance , and combining these with data from the literature we find a detection rate of 43 \pm 14 % , comparable to the detection rate for nearby radio galaxies , suggesting that group-dominant ellipticals may be more likely to contain molecular gas than their non-central counterparts . Those group-dominant galaxies which are detected typically contain \sim 2 \times 10 ^ { 8 } M _ { \odot } of molecular gas , and although most have low star formation rates ( < 1 M _ { \odot } yr ^ { -1 } ) they have short depletion times , indicating that the gas must be replenished on timescales \sim 10 ^ { 8 } yr . Almost all of the galaxies contain active nuclei , and we note while the data suggest that CO may be more common in the most radio-loud galaxies , the mass of molecular gas required to power the active nuclei through accretion is small compared to the masses observed . We consider possible origin mechanisms for the gas , through cooling of stellar ejecta within the galaxies , group-scale cooling flows , and gas-rich mergers , and find probable examples of each type within our sample , confirming that a variety of processes act to drive the build up of molecular gas in group-dominant ellipticals .