Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| isoclinism | isoclinic class of #72235323 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini | C26 | #11757162 |
| Fitting, cosocle | #72234510 | A5 |
| socle | C24 | #72209603 |
| LCS | G | |
| UCS | C1 | |
1235 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 1235 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 320 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 320 points:
(1,38,225,81,252,246)(2,251,117,209,307,82)(3,196,216,158,115,157)(4,85,72,75,166,19)(5,180,121,208,272,184)(6,138,296,102,303,129)(7,71,316,109,91,306)(8,123,265,35,33,256)(9,61,122,241,170,79)(10,311,89,195,312,220)(11,249,155,111,299,151)(12,110,200,55,203,133)(13,94,244,275,202,41)(14,190,120,276,30,224)(15,47,164,74,291,176)(16,80,78,101,197,174)(17,281,201,59,285,64)(18,21,189,77,191,95)(20,277,104,217,309,36)(22,254,279,218,263,298)(23,278,45,267,253,143)(24,124,243,165,240,90)(25,70,67,152,37,69)(26,234,181,28,266,290)(27,231,206,148,280,238)(29,318,113,194,319,154)(31,39,304,116,274,315)(32,119,250,132,269,270)(34,259,57,65,295,168)(40,50,83,257,167,207)(42,44,317,144,100,282)(43,125,313,147,134,300)(46,114,314,221,173,128)(48,302,105,287,308,150)(49,320,142,141,260,88)(51,178,261,96,273,187)(52,204,289,163,185,106)(53,214,118,98,127,237)(54,301,236,103,293,242)(56,210,193,159,247,63)(58,149,183,62,112,228)(60,186,145,99,215,245)(66,230,92,239,107,160)(68,284,229,162,153,182)(73,297,286,175,223,305)(76,222,108,188,262,288)(84,292,227,205,179,283)(86,211)(87,136,294,140,131,213)(93,146,233,171,126,156)(97,137,264,135,161,199)(130,268,169,258,235,219)(139,248,212,198,310,192)(172,232,271,177,255,226);(1,186,298,58,310,148,196,279,62,249)(2,232,78,237,116,209,255,189,118,221)(3,278,205,30,121,170,253,73,74,242)(4,133,143,267,263,69,230,45,134,38)(5,66,261,171,316,208,126,227,12,302)(6,146,305,92,315,102,239,268,56,128)(7,141,222,19,9,150,49,262,37,241)(8,111,273,21,295,82,29,223,80,90)(10,231,67,156,220,217,266,75,193,89)(11,178,85,259,142,194,297,25,243,42)(13,251,162,94,271,165,265,172,202,64)(14,248,26,214,290,276,299,218,122,181)(15,296,53,313,96,190,213,99,300,169)(16,63,147,195,280,18,233,43,20,234)(17,207,226,24,320,76,83,201,275,282)(22,115,246,163,270,28,215,225,52,318)(23,254,77,107,36,125,252,101,200,104)(27,79,238,176,319,81,127,206,164,250)(31,33,281,72,216,46,123,285,70,157)(32,71,138,135,106,113,39,272,97,291)(34,182,57,317,177,65,288,167,260,68)(35,119,258,152,40,117,198,292,166,244)(41,144,269,130,197,257,88,139,179,191)(44,284,95,245,105,100,153,174,145,109)(47,212,91,303,161,289,151,274,180,137)(48,131,264,120,192,114,301,199,183,155)(50,256,59,124,229,168,307,188,240,108)(51,149,236,158,312,219,204,184,61,311)(54,110,283,55,314,103,247,187,93,304)(60,309,84,112,129,98,277,175,185,294)(86,228,154,287,136,211,224,132,173,293)(87,210,235,159,308,140,203,286,160,306)31 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,38,225,81,252,246)(2,251,117,209,307,82)(3,196,216,158,115,157)(4,85,72,75,166,19)(5,180,121,208,272,184)(6,138,296,102,303,129)(7,71,316,109,91,306)(8,123,265,35,33,256)(9,61,122,241,170,79)(10,311,89,195,312,220)(11,249,155,111,299,151)(12,110,200,55,203,133)(13,94,244,275,202,41)(14,190,120,276,30,224)(15,47,164,74,291,176)(16,80,78,101,197,174)(17,281,201,59,285,64)(18,21,189,77,191,95)(20,277,104,217,309,36)(22,254,279,218,263,298)(23,278,45,267,253,143)(24,124,243,165,240,90)(25,70,67,152,37,69)(26,234,181,28,266,290)(27,231,206,148,280,238)(29,318,113,194,319,154)(31,39,304,116,274,315)(32,119,250,132,269,270)(34,259,57,65,295,168)(40,50,83,257,167,207)(42,44,317,144,100,282)(43,125,313,147,134,300)(46,114,314,221,173,128)(48,302,105,287,308,150)(49,320,142,141,260,88)(51,178,261,96,273,187)(52,204,289,163,185,106)(53,214,118,98,127,237)(54,301,236,103,293,242)(56,210,193,159,247,63)(58,149,183,62,112,228)(60,186,145,99,215,245)(66,230,92,239,107,160)(68,284,229,162,153,182)(73,297,286,175,223,305)(76,222,108,188,262,288)(84,292,227,205,179,283)(86,211)(87,136,294,140,131,213)(93,146,233,171,126,156)(97,137,264,135,161,199)(130,268,169,258,235,219)(139,248,212,198,310,192)(172,232,271,177,255,226), (1,186,298,58,310,148,196,279,62,249)(2,232,78,237,116,209,255,189,118,221)(3,278,205,30,121,170,253,73,74,242)(4,133,143,267,263,69,230,45,134,38)(5,66,261,171,316,208,126,227,12,302)(6,146,305,92,315,102,239,268,56,128)(7,141,222,19,9,150,49,262,37,241)(8,111,273,21,295,82,29,223,80,90)(10,231,67,156,220,217,266,75,193,89)(11,178,85,259,142,194,297,25,243,42)(13,251,162,94,271,165,265,172,202,64)(14,248,26,214,290,276,299,218,122,181)(15,296,53,313,96,190,213,99,300,169)(16,63,147,195,280,18,233,43,20,234)(17,207,226,24,320,76,83,201,275,282)(22,115,246,163,270,28,215,225,52,318)(23,254,77,107,36,125,252,101,200,104)(27,79,238,176,319,81,127,206,164,250)(31,33,281,72,216,46,123,285,70,157)(32,71,138,135,106,113,39,272,97,291)(34,182,57,317,177,65,288,167,260,68)(35,119,258,152,40,117,198,292,166,244)(41,144,269,130,197,257,88,139,179,191)(44,284,95,245,105,100,153,174,145,109)(47,212,91,303,161,289,151,274,180,137)(48,131,264,120,192,114,301,199,183,155)(50,256,59,124,229,168,307,188,240,108)(51,149,236,158,312,219,204,184,61,311)(54,110,283,55,314,103,247,187,93,304)(60,309,84,112,129,98,277,175,185,294)(86,228,154,287,136,211,224,132,173,293)(87,210,235,159,308,140,203,286,160,306));