A000679 Number of groups of order 2^n.

1, 1, 2, 5, 14, 51, 267, 2328, 56092, 10494213, 49487367289

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Examples

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 51*x^5 + 267*x^6 + 2328*x^7 + ...
		

References

• James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
• M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
• M. F. Newman, Groups of prime-power order (1990). In Groups—Canberra 1989 (pp. 49-62). Springer, Berlin, Heidelberg. See Table 1.
• M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128, Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. Applebaum, J. Clikeman, J. A. Davis, J. F. Dillon, J. Jedwab, T. Rabbani, K. Smith, and W. Yolland, Constructions of difference sets in nonabelian 2-groups, Alg. Num. Theor. (2023) Vol. 17, No. 2. 359-396. See p. 396.
• Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
• Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
• Hans Ulrich Besche, Bettina Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
• David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.
• Hans Ulrich Besche, The Small Groups library
• Bettina Eick and E. A. O'Brien, Enumerating p-groups. Group theory. J. Austral. Math. Soc. Ser. A 67 (1999), no. 2, 191-205.
• Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
• Rodney James and John Cannon, Computation of isomorphism classes of p-groups, Mathematics of Computation 23.105 (1969): 135-140.
• Rodney James, M. F. Newman, and E. A. O'Brien, The Groups of Order 128, J. Algebra 129, 136-158, 1990.
• G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
• E. A. O'Brien, The Groups of Order 256 J. Algebra 143, 219-235, 1991.
• Eugene Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142.
• Eric Weisstein's World of Mathematics, Finite Group.
• Marcel Wild, The groups of order 16 made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
• Index entries for sequences related to groups.

Crossrefs

Cf. A000001, A046058.

Programs

GAP

A000679 := List([0..8],n -> NumberSmallGroups(2^n)); # Muniru A Asiru, Oct 15 2017

Maple

seq(GroupTheory:--NumGroups(2^n),n=0..10); # Robert Israel, Oct 15 2017

Mathematica

Join[{1}, FiniteGroupCount[2^Range[10]]] (* Vincenzo Librandi, Mar 28 2018 *)

Formula

a(n) = 2^((2/27)n^3 + O(n^(8/3))).

a(n) = A000001(2^n). - Amiram Eldar, Mar 10 2024

Extensions

a(9) and a(10) found by Eamonn O'Brien

a(10) corrected by David Burrell, Jun 06 2022