# Group 512.1894 downloaded from the LMFDB on 26 October 2025. ## Various presentations of this group are stored in this file: # GPC is polycyclic presentation GPerm is permutation group # GLZ, GLFp, GLZA, GLZq, GLFq if they exist are matrix groups # Many characteristics of the group are stored as booleans in a record: # Agroup, Zgroup, abelian, almost_simple,cyclic, metabelian, # metacyclic, monomial, nilpotent, perfect, quasisimple, rational, # solvable, supersolvable # The character table is stored as a record chartbl_n_i where n is the order # of the group and i is which group of that order it is. The record is # converted to a character table using ConvertToLibraryCharacterTableNC # Constructions GPC := PcGroupCode(822956296728586764664793120872032789327267849220852593689329349,512); a := GPC.1; b := GPC.3; c := GPC.5; d := GPC.6; e := GPC.7; GPerm := Group( (1,21,5,17)(2,22,6,18)(3,23,7,19)(4,24,8,20)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28), (1,5,4,8,2,6,3,7)(9,13,11,15,10,14,12,16)(17,29,19,31,18,30,20,32)(21,27,23,26,22,28,24,25), (1,10,2,9)(3,12,4,11)(5,13)(6,14)(7,15)(8,16)(17,28,18,27)(19,25,20,26)(21,31)(22,32)(23,30)(24,29), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,4,2,3)(5,8,6,7)(9,11,10,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)(29,31,30,32), (1,2)(3,4)(13,14)(15,16)(17,18)(19,20)(29,30)(31,32), (1,2)(3,4)(5,6)(7,8)(17,18)(19,20)(21,22)(23,24), (17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32) ); # Booleans booleans_512_1894 := rec( Agroup := false, Zgroup := false, abelian := false, almost_simple := false, cyclic := false, metabelian := false, metacyclic := false, monomial := true, nilpotent := true, perfect := false, quasisimple := false, rational := false, solvable := true, supersolvable := true); # Character Table chartbl_512_1894:=rec(); chartbl_512_1894.IsFinite:= true; chartbl_512_1894.UnderlyingCharacteristic:= 0; chartbl_512_1894.UnderlyingGroup:= GPC; chartbl_512_1894.Size:= 512; chartbl_512_1894.InfoText:= "Character table for group 512.1894 downloaded from the LMFDB."; chartbl_512_1894.Identifier:= " (C2^2*Q8).Q16 "; chartbl_512_1894.NrConjugacyClasses:= 29; chartbl_512_1894.ConjugacyClasses:= [ of ..., f9, f6, f4*f5, f4*f6, f5, f2*f4*f6*f8*f9, f2*f6, f4*f5*f6*f8*f9, f4*f5*f6*f8, f5*f8, f2*f8, f2*f7, f2*f7*f9, f2*f5*f7, f2*f5*f7*f9, f2*f3, f2*f3*f5, f3*f6*f8*f9, f1*f3*f5, f1*f2*f3*f6*f8*f9, f1*f4*f6*f9, f1*f2*f4, f3*f4*f5*f7*f8*f9, f3*f7*f9, f1*f2*f4*f6*f7, f1*f3*f6*f7*f8, f1*f3*f5*f7*f8, f1*f2*f4*f5*f7]; chartbl_512_1894.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]; chartbl_512_1894.ComputedPowerMaps:= [ , [1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 5, 5, 5, 5, 6, 6, 5, 7, 7, 8, 8, 9, 10, 17, 18, 18, 17]]; chartbl_512_1894.SizesCentralizers:= [512, 512, 256, 128, 128, 128, 64, 64, 128, 128, 64, 32, 32, 32, 32, 32, 32, 32, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16]; chartbl_512_1894.ClassNames:= ["1A", "2A", "2B", "2C", "2D", "2E", "2F", "2G", "4A1", "4A-1", "4B", "4C", "4D", "4E", "4F1", "4F-1", "4G1", "4G-1", "4H", "4I1", "4I-1", "4J1", "4J-1", "8A1", "8A-1", "8B1", "8B-1", "8B3", "8B-3"]; chartbl_512_1894.OrderClassRepresentatives:= [1, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8]; chartbl_512_1894.Irr:= [[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, -1, -1, 1, -1, 1, -1, -1, -1], [1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1*E(4), -1*E(4), 1, E(4), E(4), 1, -1*E(4), 1, E(4), -1*E(4), E(4)], [1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, E(4), E(4), 1, -1*E(4), -1*E(4), 1, E(4), 1, -1*E(4), E(4), -1*E(4)], [1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1*E(4), -1*E(4), 1, E(4), E(4), -1, E(4), -1, -1*E(4), E(4), -1*E(4)], [1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, E(4), E(4), 1, -1*E(4), -1*E(4), -1, -1*E(4), -1, E(4), -1*E(4), E(4)], [2, 2, 2, 2, 2, 2, -2, -2, 2, 2, 2, 0, 0, 2, -2, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, -2, 2, -2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1*E(8)-E(8)^3, 0, E(8)+E(8)^3, E(8)+E(8)^3, -1*E(8)-E(8)^3], [2, 2, 2, 2, -2, -2, 2, 2, -2, -2, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(8)+E(8)^3, 0, -1*E(8)-E(8)^3, -1*E(8)-E(8)^3, E(8)+E(8)^3], [2, 2, 2, 2, -2, -2, 0, 0, 2, 2, -2, 0, 0, -2*E(4), 0, 2*E(4), 0, 0, -1+E(4), 1-E(4), 0, 1+E(4), -1-E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 2, 2, -2, 0, 0, 2*E(4), 0, -2*E(4), 0, 0, -1-E(4), 1+E(4), 0, 1-E(4), -1+E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 2, 2, -2, 0, 0, -2*E(4), 0, 2*E(4), 0, 0, 1-E(4), -1+E(4), 0, -1-E(4), 1+E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, 0, 0, 2, 2, -2, 0, 0, 2*E(4), 0, -2*E(4), 0, 0, 1+E(4), -1-E(4), 0, -1+E(4), 1-E(4), 0, 0, 0, 0, 0, 0], [2, 2, 2, 2, -2, -2, -2, -2, -2, -2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1*E(8)-E(8)^-1, 0, -1*E(8)-E(8)^-1, E(8)+E(8)^-1, E(8)+E(8)^-1], [2, 2, 2, 2, -2, -2, -2, -2, -2, -2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, E(8)+E(8)^-1, 0, E(8)+E(8)^-1, -1*E(8)-E(8)^-1, -1*E(8)-E(8)^-1], [4, 4, 4, 4, 4, 4, 0, 0, -4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, -4, 4, -4, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, -4, 4, -4, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 4, 4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2*E(4), 0, 2*E(4), 0, 0, 0], [4, 4, 4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2*E(4), 0, -2*E(4), 0, 0, 0], [8, 8, -8, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 8, -8, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 0, 0, 0, 0, 0, -4*E(4), 4*E(4), 0, -2, -2*E(4), 0, 0, 0, 2, 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 0, 0, 0, 0, 0, 4*E(4), -4*E(4), 0, -2, 2*E(4), 0, 0, 0, 2, -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 0, 0, 0, 0, 0, -4*E(4), 4*E(4), 0, 2, 2*E(4), 0, 0, 0, -2, -2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -8, 0, 0, 0, 0, 0, 0, 4*E(4), -4*E(4), 0, 2, -2*E(4), 0, 0, 0, -2, 2*E(4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; ConvertToLibraryCharacterTableNC(chartbl_512_1894);