LMFDB - Abstract group search results

The results below are complete, since the LMFDB contains all groups of order at most 2000 except orders larger than 500 that are multiples of 128

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Order	Exponent	Nilpotency class	Nilpotent

Center	Commutator	Central quotient	Abelianization

Abelian	Cyclic	Solvable	Simple

Perfect	Direct product	Semidirect product	Wreath product

Automorphism group	Automorphisms	Min. transitive degree	Min. permutation degree

Number of subgroups	Num. of normal subs	Num. of conj. classes

Order statistics	Family	Order factorization	Commutator length

Min. degree of $\C$-irrep	Min. degree of $\Q$-irrep	Min. degree of $\C$-rep	Min. degree of $\Q$-rep

Outer aut. group	Outer automorphisms	Metabelian	Metacyclic

Almost simple	Quasisimple	A-group	Z-group

Frattini subgroup	Derived length	Rank	Schur multiplier

Supersolvable	Monomial	Rational

Num. of char. subgroups	Num. of aut. classes	Number of divisions

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Results (1-50 of 56092 matches)

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Label	Name	Order	Exponent	Nilp. class	Der. length	Comp. length	Rank	$\card{\mathrm{conj}(G)}$	Subgroups	Subgroup classes	Normal subgroups	Center	Central quotient	Commutator	Abelianization	$\card{\mathrm{Aut}(G)}$	$\card{\mathrm{Out}(G)}$	Tr. deg	Perm. deg	$\C$-irrep deg	$\R$-irrep deg	$\Q$-irrep deg	$\C$-rep deg	$\R$-rep deg	$\Q$-rep deg	Type - length
256.1	$C_{256}$	$2^{8}$	$2^{8}$	$1$	$1$	$8$	$1$	256	$9$	$9$	$9$	$C_{256}$	$C_1$	$C_1$	$C_{256}$	$2^{7}$	$2^{7}$	$256$	$256$	$1$	$2$	$128$	$1$	$2$	$128$	Cyclic
256.2	$C_4.C_8^2$	$2^{8}$	$2^{3}$	$2$	$2$	$8$	$2$	88	$307$	$169$	$87$	$C_2^2\times C_4$	$C_4^2$	$C_4$	$C_{8}^{2}$	$2^{13} \cdot 3$	$2^{9} \cdot 3$	$256$	$32$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
256.3	$C_4.C_8^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	112	$339$	$207$	$115$	$C_2\times C_4^2$	$D_4$	$C_4$	$C_{8}^{2}$	$2^{13}$	$2^{10}$	$256$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.4	$C_2^2.C_8^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	88	$491$	$237$	$95$	$C_2^2\times C_4$	$C_2^2:C_4$	$C_2^2$	$C_{8}^{2}$	$2^{13}$	$2^{9}$	$128$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.5	$(C_2\times C_4^2):C_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	64	$651$	$261$	$67$	$C_2^3$	$C_2.C_4^2$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{13}$	$2^{8}$	$64$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.6	$C_4^2.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	64	$307$	$147$	$71$	$C_2^3$	$C_2.C_4^2$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{13}$	$2^{8}$	$256$	$32$	$-$	$-$	$-$	$7$	$12$	$16$	Nilpotent - 3
256.7	$C_4^3.C_2^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	64	$307$	$153$	$59$	$C_2^3$	$C_2.C_4^2$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{13}$	$2^{8}$	$256$	$32$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.8	$(C_2^2\times C_8):C_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	64	$491$	$215$	$79$	$C_2^3$	$C_2.C_4^2$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{13}$	$2^{8}$	$128$	$24$	$-$	$-$	$-$	$7$	$8$	$12$	Nilpotent - 3
256.9	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	64	$443$	$190$	$61$	$C_2^3$	$C_2.C_4^2$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{7}$	$128$	$24$	$-$	$-$	$-$	$7$	$10$	$12$	Nilpotent - 3
256.10	$C_4^2.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	64	$323$	$160$	$73$	$C_2^3$	$C_2.C_4^2$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{7}$	$256$	$24$	$-$	$-$	$-$	$7$	$8$	$12$	Nilpotent - 3
256.11	$C_4^3.C_2^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	64	$307$	$150$	$61$	$C_2^3$	$C_2.C_4^2$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{7}$	$256$	$24$	$-$	$-$	$-$	$7$	$10$	$12$	Nilpotent - 3
256.12	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	64	$619$	$234$	$73$	$C_2^3$	$C_2.C_4^2$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{7}$	$64$	$24$	$-$	$-$	$-$	$7$	$8$	$12$	Nilpotent - 3
256.13	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	76	$747$	$311$	$119$	$C_2^4$	$C_2^2:C_4$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{13}$	$2^{9}$	$256$	$28$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.14	$(C_2\times \OD_{16}):C_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	76	$491$	$221$	$71$	$C_2^2\times C_4$	$C_2^2:C_4$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{13}$	$2^{9}$	$128$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.15	$C_4^2.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	76	$323$	$169$	$75$	$C_2^2\times C_4$	$C_2^2:C_4$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{8}$	$256$	$24$	$-$	$-$	$-$	$5$	$8$	$12$	Nilpotent - 3
256.16	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	76	$667$	$280$	$81$	$C_2^2\times C_4$	$C_2^2:C_4$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{8}$	$64$	$24$	$-$	$-$	$-$	$5$	$8$	$12$	Nilpotent - 3
256.17	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	76	$491$	$224$	$81$	$C_2^2\times C_4$	$C_2^2:C_4$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{8}$	$128$	$24$	$-$	$-$	$-$	$5$	$8$	$12$	Nilpotent - 3
256.18	$C_4^3.C_2^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	76	$323$	$172$	$69$	$C_2^2\times C_4$	$C_2^2:C_4$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{8}$	$256$	$24$	$-$	$-$	$-$	$5$	$10$	$12$	Nilpotent - 3
256.19	$C_4^2.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	76	$339$	$176$	$93$	$C_2^2\times C_4$	$C_2^2:C_4$	$C_2\times C_4$	$C_{4} \times C_{8}$	$2^{12}$	$2^{8}$	$256$	$24$	$-$	$-$	$-$	$5$	$6$	$12$	Nilpotent - 3
256.20	$C_2^4.C_4^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$1115$	$365$	$67$	$C_2^3$	$C_2.C_4^2$	$C_2^3$	$C_{4} \times C_{8}$	$2^{13}$	$2^{8}$	$64$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.21	$C_2^4.C_4^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$443$	$181$	$59$	$C_2^3$	$C_2.C_4^2$	$C_2^3$	$C_{4} \times C_{8}$	$2^{12}$	$2^{7}$	$128$	$24$	$-$	$-$	$-$	$9$	$10$	$12$	Nilpotent - 3
256.22	$C_2^4.C_4^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$491$	$203$	$67$	$C_2^3$	$C_2.C_4^2$	$C_2^3$	$C_{4} \times C_{8}$	$2^{13}$	$2^{8}$	$128$	$24$	$-$	$-$	$-$	$9$	$10$	$12$	Nilpotent - 3
256.23	$C_2^4.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$1323$	$422$	$69$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{13}$	$2^{8}$	$64$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.24	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$619$	$229$	$55$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{12}$	$2^{7}$	$64$	$24$	$-$	$-$	$-$	$6$	$12$	$12$	Nilpotent - 3
256.25	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$651$	$238$	$61$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{12}$	$2^{7}$	$64$	$24$	$-$	$-$	$-$	$6$	$10$	$12$	Nilpotent - 3
256.26	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$443$	$179$	$51$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{12}$	$2^{7}$	$128$	$24$	$-$	$-$	$-$	$6$	$12$	$12$	Nilpotent - 3
256.27	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$667$	$245$	$75$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{12}$	$2^{7}$	$64$	$24$	$-$	$-$	$-$	$6$	$8$	$12$	Nilpotent - 3
256.28	$C_4^2.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$323$	$146$	$69$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{13}$	$2^{8}$	$256$	$32$	$-$	$-$	$-$	$6$	$8$	$16$	Nilpotent - 3
256.29	$C_4^2.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$307$	$139$	$55$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{12}$	$2^{7}$	$256$	$32$	$-$	$-$	$-$	$6$	$10$	$16$	Nilpotent - 3
256.30	$C_4^3.C_2^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$323$	$152$	$57$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{13}$	$2^{8}$	$256$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.31	$C_4^3.C_2^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$307$	$142$	$49$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{12}$	$2^{7}$	$256$	$24$	$-$	$-$	$-$	$6$	$12$	$12$	Nilpotent - 3
256.32	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$443$	$176$	$57$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{13}$	$2^{8}$	$128$	$32$	$-$	$-$	$-$	$6$	$10$	$16$	Nilpotent - 3
256.33	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$491$	$201$	$63$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{12}$	$2^{7}$	$128$	$24$	$-$	$-$	$-$	$6$	$10$	$12$	Nilpotent - 3
256.34	$(C_2\times C_4^2).D_4$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$443$	$182$	$49$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{13}$	$2^{8}$	$128$	$32$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.35	$C_2^5.Q_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$907$	$304$	$53$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{13} \cdot 3$	$2^{8} \cdot 3$	$64$	$32$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.36	$C_4^3.C_2^2$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$307$	$145$	$47$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{12} \cdot 3$	$2^{7} \cdot 3$	$256$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.37	$C_4^2.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$323$	$152$	$61$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{13}$	$2^{8}$	$256$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.38	$C_4^2.D_8$	$2^{8}$	$2^{3}$	$3$	$2$	$8$	$2$	52	$355$	$166$	$89$	$C_2^3$	$C_2.C_4^2$	$C_2^2\times C_4$	$C_{4}^{2}$	$2^{13} \cdot 3$	$2^{8} \cdot 3$	$256$	$24$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
256.39	$C_{16}^2$	$2^{8}$	$2^{4}$	$1$	$1$	$8$	$2$	256	$83$	$83$	$83$	$C_{16}^2$	$C_1$	$C_1$	$C_{16}^{2}$	$2^{13} \cdot 3$	$2^{13} \cdot 3$	$256$	$32$	$-$	$-$	$-$	$2$	$4$	$16$	Abelian - 2
256.40	$C_{16}:C_{16}$	$2^{8}$	$2^{4}$	$2$	$2$	$8$	$2$	160	$83$	$75$	$67$	$C_8^2$	$C_2^2$	$C_2$	$C_{8} \times C_{16}$	$2^{13}$	$2^{11}$	$256$	$32$	$-$	$-$	$-$	$3$	$6$	$16$	Nilpotent - 2
256.41	$C_{16}:C_{16}$	$2^{8}$	$2^{4}$	$2$	$2$	$8$	$2$	88	$83$	$61$	$47$	$C_4^2$	$C_4^2$	$C_4$	$C_{4} \times C_{16}$	$2^{11}$	$2^{7}$	$256$	$32$	$-$	$-$	$-$	$5$	$10$	$16$	Nilpotent - 2
256.42	$C_2^4:C_{16}$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	52	$787$	$238$	$31$	$C_2\times C_4$	$C_2^3:C_4$	$C_2^3$	$C_{2} \times C_{16}$	$2^{11}$	$2^{6}$	$32$	$24$	$-$	$-$	$-$	$5$	$6$	$12$	Nilpotent - 4
256.43	$C_2^3.\OD_{32}$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	52	$339$	$120$	$31$	$C_2\times C_4$	$C_2^3:C_4$	$C_2^3$	$C_{2} \times C_{16}$	$2^{11}$	$2^{6}$	$64$	$32$	$-$	$-$	$-$	$5$	$10$	$16$	Nilpotent - 4
256.44	$(C_2\times D_4):C_{16}$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	52	$395$	$124$	$31$	$C_2\times C_4$	$C_2^3:C_4$	$C_2\times C_4$	$C_{2} \times C_{16}$	$2^{11}$	$2^{6}$	$64$	$24$	$-$	$-$	$-$	$5$	$6$	$12$	Nilpotent - 4
256.45	$(C_2\times Q_8):C_{16}$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	52	$251$	$98$	$31$	$C_2\times C_4$	$C_2^3:C_4$	$C_2\times C_4$	$C_{2} \times C_{16}$	$2^{11}$	$2^{6}$	$128$	$32$	$-$	$-$	$-$	$5$	$10$	$16$	Nilpotent - 4
256.46	$C_2^4.\OD_{16}$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	46	$611$	$177$	$27$	$C_2\times C_4$	$C_2^3:C_4$	$C_2^2\times C_4$	$C_{2} \times C_{8}$	$2^{11}$	$2^{6}$	$64$	$40$	$-$	$-$	$-$	$6$	$8$	$20$	Nilpotent - 4
256.47	$C_2^4.\OD_{16}$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	46	$291$	$97$	$27$	$C_2\times C_4$	$C_2^3:C_4$	$C_2^2\times C_4$	$C_{2} \times C_{8}$	$2^{11}$	$2^{6}$	$64$	$48$	$-$	$-$	$-$	$6$	$12$	$24$	Nilpotent - 4
256.48	$(C_2^2\times C_8).D_4$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	46	$395$	$120$	$27$	$C_2\times C_4$	$C_2^3:C_4$	$C_2^2\times C_4$	$C_{2} \times C_{8}$	$2^{11}$	$2^{6}$	$64$	$24$	$-$	$-$	$-$	$6$	$8$	$12$	Nilpotent - 4
256.49	$(C_2^2\times C_8).D_4$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	46	$251$	$94$	$27$	$C_2\times C_4$	$C_2^3:C_4$	$C_2^2\times C_4$	$C_{2} \times C_{8}$	$2^{11}$	$2^{6}$	$128$	$32$	$-$	$-$	$-$	$6$	$12$	$16$	Nilpotent - 4
256.50	$C_4^2:C_{16}$	$2^{8}$	$2^{4}$	$4$	$2$	$8$	$2$	52	$219$	$90$	$31$	$C_2\times C_4$	$C_2^3:C_4$	$C_2\times C_4$	$C_{2} \times C_{16}$	$2^{11}$	$2^{6}$	$64$	$32$	$-$	$-$	$-$	$5$	$10$	$16$	Nilpotent - 4

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