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SD₂₀₄₈ #72210182

• p-group
• nilpotent
• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

Subgroups

31 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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);(2,512)(3,1023)(4,510)(5,1021)(6,508)(7,1019)(8,506)(9,1017)(10,504)(11,1015)(12,502)(13,1013)(14,500)(15,1011)(16,498)(17,1009)(18,496)(19,1007)(20,494)(21,1005)(22,492)(23,1003)(24,490)(25,1001)(26,488)(27,999)(28,486)(29,997)(30,484)(31,995)(32,482)(33,993)(34,480)(35,991)(36,478)(37,989)(38,476)(39,987)(40,474)(41,985)(42,472)(43,983)(44,470)(45,981)(46,468)(47,979)(48,466)(49,977)(50,464)(51,975)(52,462)(53,973)(54,460)(55,971)(56,458)(57,969)(58,456)(59,967)(60,454)(61,965)(62,452)(63,963)(64,450)(65,961)(66,448)(67,959)(68,446)(69,957)(70,444)(71,955)(72,442)(73,953)(74,440)(75,951)(76,438)(77,949)(78,436)(79,947)(80,434)(81,945)(82,432)(83,943)(84,430)(85,941)(86,428)(87,939)(88,426)(89,937)(90,424)(91,935)(92,422)(93,933)(94,420)(95,931)(96,418)(97,929)(98,416)(99,927)(100,414)(101,925)(102,412)(103,923)(104,410)(105,921)(106,408)(107,919)(108,406)(109,917)(110,404)(111,915)(112,402)(113,913)(114,400)(115,911)(116,398)(117,909)(118,396)(119,907)(120,394)(121,905)(122,392)(123,903)(124,390)(125,901)(126,388)(127,899)(128,386)(129,897)(130,384)(131,895)(132,382)(133,893)(134,380)(135,891)(136,378)(137,889)(138,376)(139,887)(140,374)(141,885)(142,372)(143,883)(144,370)(145,881)(146,368)(147,879)(148,366)(149,877)(150,364)(151,875)(152,362)(153,873)(154,360)(155,871)(156,358)(157,869)(158,356)(159,867)(160,354)(161,865)(162,352)(163,863)(164,350)(165,861)(166,348)(167,859)(168,346)(169,857)(170,344)(171,855)(172,342)(173,853)(174,340)(175,851)(176,338)(177,849)(178,336)(179,847)(180,334)(181,845)(182,332)(183,843)(184,330)(185,841)(186,328)(187,839)(188,326)(189,837)(190,324)(191,835)(192,322)(193,833)(194,320)(195,831)(196,318)(197,829)(198,316)(199,827)(200,314)(201,825)(202,312)(203,823)(204,310)(205,821)(206,308)(207,819)(208,306)(209,817)(210,304)(211,815)(212,302)(213,813)(214,300)(215,811)(216,298)(217,809)(218,296)(219,807)(220,294)(221,805)(222,292)(223,803)(224,290)(225,801)(226,288)(227,799)(228,286)(229,797)(230,284)(231,795)(232,282)(233,793)(234,280)(235,791)(236,278)(237,789)(238,276)(239,787)(240,274)(241,785)(242,272)(243,783)(244,270)(245,781)(246,268)(247,779)(248,266)(249,777)(250,264)(251,775)(252,262)(253,773)(254,260)(255,771)(256,258)(257,769)(259,767)(261,765)(263,763)(265,761)(267,759)(269,757)(271,755)(273,753)(275,751)(277,749)(279,747)(281,745)(283,743)(285,741)(287,739)(289,737)(291,735)(293,733)(295,731)(297,729)(299,727)(301,725)(303,723)(305,721)(307,719)(309,717)(311,715)(313,713)(315,711)(317,709)(319,707)(321,705)(323,703)(325,701)(327,699)(329,697)(331,695)(333,693)(335,691)(337,689)(339,687)(341,685)(343,683)(345,681)(347,679)(349,677)(351,675)(353,673)(355,671)(357,669)(359,667)(361,665)(363,663)(365,661)(367,659)(369,657)(371,655)(373,653)(375,651)(377,649)(379,647)(381,645)(383,643)(385,641)(387,639)(389,637)(391,635)(393,633)(395,631)(397,629)(399,627)(401,625)(403,623)(405,621)(407,619)(409,617)(411,615)(413,613)(415,611)(417,609)(419,607)(421,605)(423,603)(425,601)(427,599)(429,597)(431,595)(433,593)(435,591)(437,589)(439,587)(441,585)(443,583)(445,581)(447,579)(449,577)(451,575)(453,573)(455,571)(457,569)(459,567)(461,565)(463,563)(465,561)(467,559)(469,557)(471,555)(473,553)(475,551)(477,549)(479,547)(481,545)(483,543)(485,541)(487,539)(489,537)(491,535)(493,533)(495,531)(497,529)(499,527)(501,525)(503,523)(505,521)(507,519)(509,517)(511,515)(514,1024)(516,1022)(518,1020)(520,1018)(522,1016)(524,1014)(526,1012)(528,1010)(530,1008)(532,1006)(534,1004)(536,1002)(538,1000)(540,998)(542,996)(544,994)(546,992)(548,990)(550,988)(552,986)(554,984)(556,982)(558,980)(560,978)(562,976)(564,974)(566,972)(568,970)(570,968)(572,966)(574,964)(576,962)(578,960)(580,958)(582,956)(584,954)(586,952)(588,950)(590,948)(592,946)(594,944)(596,942)(598,940)(600,938)(602,936)(604,934)(606,932)(608,930)(610,928)(612,926)(614,924)(616,922)(618,920)(620,918)(622,916)(624,914)(626,912)(628,910)(630,908)(632,906)(634,904)(636,902)(638,900)(640,898)(642,896)(644,894)(646,892)(648,890)(650,888)(652,886)(654,884)(656,882)(658,880)(660,878)(662,876)(664,874)(666,872)(668,870)(670,868)(672,866)(674,864)(676,862)(678,860)(680,858)(682,856)(684,854)(686,852)(688,850)(690,848)(692,846)(694,844)(696,842)(698,840)(700,838)(702,836)(704,834)(706,832)(708,830)(710,828)(712,826)(714,824)(716,822)(718,820)(720,818)(722,816)(724,814)(726,812)(728,810)(730,808)(732,806)(734,804)(736,802)(738,800)(740,798)(742,796)(744,794)(746,792)(748,790)(750,788)(752,786)(754,784)(756,782)(758,780)(760,778)(762,776)(764,774)(766,772)(768,770)

Polycyclic presentation:

2^11|b^2=cdk,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=ij,i^2=jk,j^2=k,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,h^a=hi,i^a=ij,j^a=jk

Conjugacy classes & character table

515 conjugacy classes — view the full character table.

GAP

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*F.4*F.11);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);