Finite Group DatabasebetaFinite Group Databasebeta| order | 32128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic8032 |
| Sylow | C251 ⋊ Q128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8032 |
| commutator, cosocle | C32×C251 | C22 |
| Frattini | C32 | C2×D251 |
| Fitting | C64×C251 | C2 |
| socle, prime radical | C2×C251 | D32 |
| hypercenter | C64 | D251 |
| hypocenter | C251 | Q128 |
| LCS | G ▷ C32×C251 ▷ C16×C251 ▷ C8×C251 ▷ C4×C251 ▷ C2×C251 ▷ C251 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 379 |
| transitive degree | μt(G) = 32128 |
A minimal faithful permutation representation — 2 generators on 379 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)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(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);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,379)(131,378)(132,377)(133,376)(134,375)(135,374)(136,373)(137,372)(138,371)(139,370)(140,369)(141,368)(142,367)(143,366)(144,365)(145,364)(146,363)(147,362)(148,361)(149,360)(150,359)(151,358)(152,357)(153,356)(154,355)(155,354)(156,353)(157,352)(158,351)(159,350)(160,349)(161,348)(162,347)(163,346)(164,345)(165,344)(166,343)(167,342)(168,341)(169,340)(170,339)(171,338)(172,337)(173,336)(174,335)(175,334)(176,333)(177,332)(178,331)(179,330)(180,329)(181,328)(182,327)(183,326)(184,325)(185,324)(186,323)(187,322)(188,321)(189,320)(190,319)(191,318)(192,317)(193,316)(194,315)(195,314)(196,313)(197,312)(198,311)(199,310)(200,309)(201,308)(202,307)(203,306)(204,305)(205,304)(206,303)(207,302)(208,301)(209,300)(210,299)(211,298)(212,297)(213,296)(214,295)(215,294)(216,293)(217,292)(218,291)(219,290)(220,289)(221,288)(222,287)(223,286)(224,285)(225,284)(226,283)(227,282)(228,281)(229,280)(230,279)(231,278)(232,277)(233,276)(234,275)(235,274)(236,273)(237,272)(238,271)(239,270)(240,269)(241,268)(242,267)(243,266)(244,265)(245,264)(246,263)(247,262)(248,261)(249,260)(250,259)(251,258)(252,257)(253,256)(254,255)Polycyclic presentation:
2^7,251|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^250,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^2508035 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32128, 2728);