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D1249 #14578778
Aliases
Splittings
Subgroups
4 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 1249 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249);(2,1249)(3,1248)(4,1247)(5,1246)(6,1245)(7,1244)(8,1243)(9,1242)(10,1241)(11,1240)(12,1239)(13,1238)(14,1237)(15,1236)(16,1235)(17,1234)(18,1233)(19,1232)(20,1231)(21,1230)(22,1229)(23,1228)(24,1227)(25,1226)(26,1225)(27,1224)(28,1223)(29,1222)(30,1221)(31,1220)(32,1219)(33,1218)(34,1217)(35,1216)(36,1215)(37,1214)(38,1213)(39,1212)(40,1211)(41,1210)(42,1209)(43,1208)(44,1207)(45,1206)(46,1205)(47,1204)(48,1203)(49,1202)(50,1201)(51,1200)(52,1199)(53,1198)(54,1197)(55,1196)(56,1195)(57,1194)(58,1193)(59,1192)(60,1191)(61,1190)(62,1189)(63,1188)(64,1187)(65,1186)(66,1185)(67,1184)(68,1183)(69,1182)(70,1181)(71,1180)(72,1179)(73,1178)(74,1177)(75,1176)(76,1175)(77,1174)(78,1173)(79,1172)(80,1171)(81,1170)(82,1169)(83,1168)(84,1167)(85,1166)(86,1165)(87,1164)(88,1163)(89,1162)(90,1161)(91,1160)(92,1159)(93,1158)(94,1157)(95,1156)(96,1155)(97,1154)(98,1153)(99,1152)(100,1151)(101,1150)(102,1149)(103,1148)(104,1147)(105,1146)(106,1145)(107,1144)(108,1143)(109,1142)(110,1141)(111,1140)(112,1139)(113,1138)(114,1137)(115,1136)(116,1135)(117,1134)(118,1133)(119,1132)(120,1131)(121,1130)(122,1129)(123,1128)(124,1127)(125,1126)(126,1125)(127,1124)(128,1123)(129,1122)(130,1121)(131,1120)(132,1119)(133,1118)(134,1117)(135,1116)(136,1115)(137,1114)(138,1113)(139,1112)(140,1111)(141,1110)(142,1109)(143,1108)(144,1107)(145,1106)(146,1105)(147,1104)(148,1103)(149,1102)(150,1101)(151,1100)(152,1099)(153,1098)(154,1097)(155,1096)(156,1095)(157,1094)(158,1093)(159,1092)(160,1091)(161,1090)(162,1089)(163,1088)(164,1087)(165,1086)(166,1085)(167,1084)(168,1083)(169,1082)(170,1081)(171,1080)(172,1079)(173,1078)(174,1077)(175,1076)(176,1075)(177,1074)(178,1073)(179,1072)(180,1071)(181,1070)(182,1069)(183,1068)(184,1067)(185,1066)(186,1065)(187,1064)(188,1063)(189,1062)(190,1061)(191,1060)(192,1059)(193,1058)(194,1057)(195,1056)(196,1055)(197,1054)(198,1053)(199,1052)(200,1051)(201,1050)(202,1049)(203,1048)(204,1047)(205,1046)(206,1045)(207,1044)(208,1043)(209,1042)(210,1041)(211,1040)(212,1039)(213,1038)(214,1037)(215,1036)(216,1035)(217,1034)(218,1033)(219,1032)(220,1031)(221,1030)(222,1029)(223,1028)(224,1027)(225,1026)(226,1025)(227,1024)(228,1023)(229,1022)(230,1021)(231,1020)(232,1019)(233,1018)(234,1017)(235,1016)(236,1015)(237,1014)(238,1013)(239,1012)(240,1011)(241,1010)(242,1009)(243,1008)(244,1007)(245,1006)(246,1005)(247,1004)(248,1003)(249,1002)(250,1001)(251,1000)(252,999)(253,998)(254,997)(255,996)(256,995)(257,994)(258,993)(259,992)(260,991)(261,990)(262,989)(263,988)(264,987)(265,986)(266,985)(267,984)(268,983)(269,982)(270,981)(271,980)(272,979)(273,978)(274,977)(275,976)(276,975)(277,974)(278,973)(279,972)(280,971)(281,970)(282,969)(283,968)(284,967)(285,966)(286,965)(287,964)(288,963)(289,962)(290,961)(291,960)(292,959)(293,958)(294,957)(295,956)(296,955)(297,954)(298,953)(299,952)(300,951)(301,950)(302,949)(303,948)(304,947)(305,946)(306,945)(307,944)(308,943)(309,942)(310,941)(311,940)(312,939)(313,938)(314,937)(315,936)(316,935)(317,934)(318,933)(319,932)(320,931)(321,930)(322,929)(323,928)(324,927)(325,926)(326,925)(327,924)(328,923)(329,922)(330,921)(331,920)(332,919)(333,918)(334,917)(335,916)(336,915)(337,914)(338,913)(339,912)(340,911)(341,910)(342,909)(343,908)(344,907)(345,906)(346,905)(347,904)(348,903)(349,902)(350,901)(351,900)(352,899)(353,898)(354,897)(355,896)(356,895)(357,894)(358,893)(359,892)(360,891)(361,890)(362,889)(363,888)(364,887)(365,886)(366,885)(367,884)(368,883)(369,882)(370,881)(371,880)(372,879)(373,878)(374,877)(375,876)(376,875)(377,874)(378,873)(379,872)(380,871)(381,870)(382,869)(383,868)(384,867)(385,866)(386,865)(387,864)(388,863)(389,862)(390,861)(391,860)(392,859)(393,858)(394,857)(395,856)(396,855)(397,854)(398,853)(399,852)(400,851)(401,850)(402,849)(403,848)(404,847)(405,846)(406,845)(407,844)(408,843)(409,842)(410,841)(411,840)(412,839)(413,838)(414,837)(415,836)(416,835)(417,834)(418,833)(419,832)(420,831)(421,830)(422,829)(423,828)(424,827)(425,826)(426,825)(427,824)(428,823)(429,822)(430,821)(431,820)(432,819)(433,818)(434,817)(435,816)(436,815)(437,814)(438,813)(439,812)(440,811)(441,810)(442,809)(443,808)(444,807)(445,806)(446,805)(447,804)(448,803)(449,802)(450,801)(451,800)(452,799)(453,798)(454,797)(455,796)(456,795)(457,794)(458,793)(459,792)(460,791)(461,790)(462,789)(463,788)(464,787)(465,786)(466,785)(467,784)(468,783)(469,782)(470,781)(471,780)(472,779)(473,778)(474,777)(475,776)(476,775)(477,774)(478,773)(479,772)(480,771)(481,770)(482,769)(483,768)(484,767)(485,766)(486,765)(487,764)(488,763)(489,762)(490,761)(491,760)(492,759)(493,758)(494,757)(495,756)(496,755)(497,754)(498,753)(499,752)(500,751)(501,750)(502,749)(503,748)(504,747)(505,746)(506,745)(507,744)(508,743)(509,742)(510,741)(511,740)(512,739)(513,738)(514,737)(515,736)(516,735)(517,734)(518,733)(519,732)(520,731)(521,730)(522,729)(523,728)(524,727)(525,726)(526,725)(527,724)(528,723)(529,722)(530,721)(531,720)(532,719)(533,718)(534,717)(535,716)(536,715)(537,714)(538,713)(539,712)(540,711)(541,710)(542,709)(543,708)(544,707)(545,706)(546,705)(547,704)(548,703)(549,702)(550,701)(551,700)(552,699)(553,698)(554,697)(555,696)(556,695)(557,694)(558,693)(559,692)(560,691)(561,690)(562,689)(563,688)(564,687)(565,686)(566,685)(567,684)(568,683)(569,682)(570,681)(571,680)(572,679)(573,678)(574,677)(575,676)(576,675)(577,674)(578,673)(579,672)(580,671)(581,670)(582,669)(583,668)(584,667)(585,666)(586,665)(587,664)(588,663)(589,662)(590,661)(591,660)(592,659)(593,658)(594,657)(595,656)(596,655)(597,654)(598,653)(599,652)(600,651)(601,650)(602,649)(603,648)(604,647)(605,646)(606,645)(607,644)(608,643)(609,642)(610,641)(611,640)(612,639)(613,638)(614,637)(615,636)(616,635)(617,634)(618,633)(619,632)(620,631)(621,630)(622,629)(623,628)(624,627)(625,626)
Polycyclic presentation:
2,1249|b^a=b^1248
Conjugacy classes & character table
626 conjugacy classes — view the full character table.
Relationships
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2498, 1);
Finite Group Databasebeta