Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C512 · Schur multiplier (H2) = C1 |
| modular maximal cyclic | M2048 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C512 | C22 |
| commutator, socle | C2 | C2×C512 |
| prime radical | C22 | C512 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C512 ◁ G | |
| LECS | G ▷ C512 ▷ C256 ▷ C128 ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
31 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1024 |
| transitive degree | μt(G) = 1024 |
A minimal faithful permutation representation — 2 generators on 1024 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024);(2,514)(4,516)(6,518)(8,520)(10,522)(12,524)(14,526)(16,528)(18,530)(20,532)(22,534)(24,536)(26,538)(28,540)(30,542)(32,544)(34,546)(36,548)(38,550)(40,552)(42,554)(44,556)(46,558)(48,560)(50,562)(52,564)(54,566)(56,568)(58,570)(60,572)(62,574)(64,576)(66,578)(68,580)(70,582)(72,584)(74,586)(76,588)(78,590)(80,592)(82,594)(84,596)(86,598)(88,600)(90,602)(92,604)(94,606)(96,608)(98,610)(100,612)(102,614)(104,616)(106,618)(108,620)(110,622)(112,624)(114,626)(116,628)(118,630)(120,632)(122,634)(124,636)(126,638)(128,640)(130,642)(132,644)(134,646)(136,648)(138,650)(140,652)(142,654)(144,656)(146,658)(148,660)(150,662)(152,664)(154,666)(156,668)(158,670)(160,672)(162,674)(164,676)(166,678)(168,680)(170,682)(172,684)(174,686)(176,688)(178,690)(180,692)(182,694)(184,696)(186,698)(188,700)(190,702)(192,704)(194,706)(196,708)(198,710)(200,712)(202,714)(204,716)(206,718)(208,720)(210,722)(212,724)(214,726)(216,728)(218,730)(220,732)(222,734)(224,736)(226,738)(228,740)(230,742)(232,744)(234,746)(236,748)(238,750)(240,752)(242,754)(244,756)(246,758)(248,760)(250,762)(252,764)(254,766)(256,768)(258,770)(260,772)(262,774)(264,776)(266,778)(268,780)(270,782)(272,784)(274,786)(276,788)(278,790)(280,792)(282,794)(284,796)(286,798)(288,800)(290,802)(292,804)(294,806)(296,808)(298,810)(300,812)(302,814)(304,816)(306,818)(308,820)(310,822)(312,824)(314,826)(316,828)(318,830)(320,832)(322,834)(324,836)(326,838)(328,840)(330,842)(332,844)(334,846)(336,848)(338,850)(340,852)(342,854)(344,856)(346,858)(348,860)(350,862)(352,864)(354,866)(356,868)(358,870)(360,872)(362,874)(364,876)(366,878)(368,880)(370,882)(372,884)(374,886)(376,888)(378,890)(380,892)(382,894)(384,896)(386,898)(388,900)(390,902)(392,904)(394,906)(396,908)(398,910)(400,912)(402,914)(404,916)(406,918)(408,920)(410,922)(412,924)(414,926)(416,928)(418,930)(420,932)(422,934)(424,936)(426,938)(428,940)(430,942)(432,944)(434,946)(436,948)(438,950)(440,952)(442,954)(444,956)(446,958)(448,960)(450,962)(452,964)(454,966)(456,968)(458,970)(460,972)(462,974)(464,976)(466,978)(468,980)(470,982)(472,984)(474,986)(476,988)(478,990)(480,992)(482,994)(484,996)(486,998)(488,1000)(490,1002)(492,1004)(494,1006)(496,1008)(498,1010)(500,1012)(502,1014)(504,1016)(506,1018)(508,1020)(510,1022)(512,1024)Polycyclic presentation:
2^11|b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,g^2=h,h^2=i,i^2=j,j^2=k,b^a=bk1280 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);