A000679 Number of groups of order 2^n. (Formerly M1470 N0581)

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

Offset

0,3

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

Table of n, a(n) for n=0..10.

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.

Formula

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000001, A046058.

Sequence in context: A049338 A306892 A115275 * A266932 A243787 A275825

Adjacent sequences: A000676 A000677 A000678 * A000680 A000681 A000682

Keyword

nonn,hard,more,nice

Author

N. J. A. Sloane

Extensions

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

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

Status

approved