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Elements of the group are displayed as words in the presentation $\langle a, b \mid b^{64}=1, a^{2}=b^{32}, b^{a}=b^{63} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P
$Q_{128}$ 1A $1$ $1$ $Q_{128}$ 1A $1$
$Q_{128}$ 2A $2$ $1$ $Q_{128}$ 1A $b^{32}$
$Q_{128}$ 4A $4$ $2$ $C_{64}$ 2A $b^{16}$
$Q_{128}$ 4B $4$ $32$ $C_4$ 2A $ab^{35}$
$Q_{128}$ 4C $4$ $32$ $C_4$ 2A $ab^{8}$
$Q_{128}$ 8A1 $8$ $2$ $C_{64}$ 4A $b^{8}$
$Q_{128}$ 8A3 $8$ $2$ $C_{64}$ 4A $b^{24}$
$Q_{128}$ 16A1 $16$ $2$ $C_{64}$ 8A1 $b^{4}$
$Q_{128}$ 16A3 $16$ $2$ $C_{64}$ 8A3 $b^{12}$
$Q_{128}$ 16A5 $16$ $2$ $C_{64}$ 8A3 $b^{20}$
$Q_{128}$ 16A7 $16$ $2$ $C_{64}$ 8A1 $b^{28}$
$Q_{128}$ 32A1 $32$ $2$ $C_{64}$ 16A1 $b^{2}$
$Q_{128}$ 32A3 $32$ $2$ $C_{64}$ 16A3 $b^{6}$
$Q_{128}$ 32A5 $32$ $2$ $C_{64}$ 16A5 $b^{10}$
$Q_{128}$ 32A7 $32$ $2$ $C_{64}$ 16A7 $b^{14}$
$Q_{128}$ 32A9 $32$ $2$ $C_{64}$ 16A7 $b^{18}$
$Q_{128}$ 32A11 $32$ $2$ $C_{64}$ 16A5 $b^{22}$
$Q_{128}$ 32A13 $32$ $2$ $C_{64}$ 16A3 $b^{26}$
$Q_{128}$ 32A15 $32$ $2$ $C_{64}$ 16A1 $b^{30}$
$Q_{128}$ 64A1 $64$ $2$ $C_{64}$ 32A1 $b$
$Q_{128}$ 64A3 $64$ $2$ $C_{64}$ 32A3 $b^{3}$
$Q_{128}$ 64A5 $64$ $2$ $C_{64}$ 32A5 $b^{5}$
$Q_{128}$ 64A7 $64$ $2$ $C_{64}$ 32A7 $b^{7}$
$Q_{128}$ 64A9 $64$ $2$ $C_{64}$ 32A9 $b^{9}$
$Q_{128}$ 64A11 $64$ $2$ $C_{64}$ 32A11 $b^{11}$
$Q_{128}$ 64A13 $64$ $2$ $C_{64}$ 32A13 $b^{13}$
$Q_{128}$ 64A15 $64$ $2$ $C_{64}$ 32A15 $b^{49}$
$Q_{128}$ 64A17 $64$ $2$ $C_{64}$ 32A15 $b^{17}$
$Q_{128}$ 64A19 $64$ $2$ $C_{64}$ 32A13 $b^{19}$
$Q_{128}$ 64A21 $64$ $2$ $C_{64}$ 32A11 $b^{21}$
$Q_{128}$ 64A23 $64$ $2$ $C_{64}$ 32A9 $b^{41}$
$Q_{128}$ 64A25 $64$ $2$ $C_{64}$ 32A7 $b^{25}$
$Q_{128}$ 64A27 $64$ $2$ $C_{64}$ 32A5 $b^{37}$
$Q_{128}$ 64A29 $64$ $2$ $C_{64}$ 32A3 $b^{35}$
$Q_{128}$ 64A31 $64$ $2$ $C_{64}$ 32A1 $b^{33}$
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