Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2 | #72227399 |
| commutator, hypocenter | G | C1 |
| Frattini | C24×C4 | #11757163 |
| Fitting, cosocle | C24×#585 | A5 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
At least 78661 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has at least 78661 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:
(1,86,285,250,18,19,176,278,307,214)(2,271,192,257,185,54,119,287,312,31)(3,14,100,300,261,55,30,177,320,289)(4,27,315,292,120,39,110,293,294,6)(5,169,318,244,74,32,168,304,226,40)(7,71,314,253,79,121,48,310,311,13)(8,263,201,88,92,23,186,204,42,38)(9,203,306,270,137,65,225,317,269,46)(10,61,166,151,21,49,136,267,258,59)(11,102,316,298,163,94,57,272,290,99)(12,75,187,231,297,95,41,209,129,302)(15,189,221,134,265,45,116,228,133,178)(16,259,227,230,91,63,150,303,131,68)(17,90,82,148,43,28,36,172,251,20)(22,260,80,78,51,62,202,165,264,101)(24,159,284,281,25,47,220,252,193,140)(26,216,295,276,107,37,208,319,175,64)(29,118,248,288,152,111,77,238,291,70)(33,89,219,164,162,83,180,283,262,98)(34,229,309,191,117,84,145,249,125,93)(35,76,104,242,81,87,167,44,286,170)(50,173,275,296,154,56,52,198,301,109)(53,161,135,254,115,274,96,234,149,114)(58,97,237,256,188,128,85,240,313,277)(60,207,123,199,124,132,155,113,138,190)(66,232,233,223,218,144,130,305,222,127)(67,183,174,308,142,160,211,236,266,241)(69,279,255,73,181,157,179,156,205,146)(72,243,299,141,200,158,143,280,239,139)(103,215,212,105,108,194,206,184,197,153)(106,126,196,171,195,217,224,268,273,235)(112,213,210,246,182,122,247,282,245,147);(1,138,262,63,36,155,19,199,164,16,90,207)(2,66,80,9,45,156,54,144,165,65,15,255)(3,91,13,46,99,64,55,68,79,137,163,107)(4,224,133,173,51,245,39,126,134,52,101,246)(5,146,115,47,61,222,32,181,114,24,136,223)(6,25,120,140)(7,72,88,34,167,141,121,158,42,84,76,239)(8,206,41,197,170,37,23,215,75,105,81,26)(10,179,292,28,305,289,49,279,294,17,233,261)(11,211,20,97,33,142,94,183,43,85,83,241)(12,111,18,117,185,128,95,29,214,93,31,58)(14,122,59,118,161,235,30,112,21,77,96,195)(22,177,309,104,57,291,62,100,249,44,102,288)(27,296,234,71,313,180,110,301,135,48,256,89)(35,280,297,274,308,307,87,299,302,53,266,250)(38,194,253,265,132,298,92,103,311,178,60,290)(40,56,74,50)(67,228,152,106,151,109,160,221,70,217,258,154)(69,248,310,108,240,320,157,238,314,153,237,300)(73,260,281,113,149,270,205,202,193,123,254,269)(78,171,226,98,282,312,264,273,244,162,210,257)(82,168,284,189,187,303,172,169,252,116,209,227)(86,317,267,271,319,263,176,306,166,119,295,186)(124,198,293,232,229,272,190,275,315,130,145,316)(125,147,148,188,200,201,191,182,251,277,139,204)(127,219,131,212,286,175,218,283,230,184,242,276)(129,143,150,287,196,225,231,243,259,192,268,203)(159,304,247,208,278,174,220,318,213,216,285,236)62 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((1,86,285,250,18,19,176,278,307,214)(2,271,192,257,185,54,119,287,312,31)(3,14,100,300,261,55,30,177,320,289)(4,27,315,292,120,39,110,293,294,6)(5,169,318,244,74,32,168,304,226,40)(7,71,314,253,79,121,48,310,311,13)(8,263,201,88,92,23,186,204,42,38)(9,203,306,270,137,65,225,317,269,46)(10,61,166,151,21,49,136,267,258,59)(11,102,316,298,163,94,57,272,290,99)(12,75,187,231,297,95,41,209,129,302)(15,189,221,134,265,45,116,228,133,178)(16,259,227,230,91,63,150,303,131,68)(17,90,82,148,43,28,36,172,251,20)(22,260,80,78,51,62,202,165,264,101)(24,159,284,281,25,47,220,252,193,140)(26,216,295,276,107,37,208,319,175,64)(29,118,248,288,152,111,77,238,291,70)(33,89,219,164,162,83,180,283,262,98)(34,229,309,191,117,84,145,249,125,93)(35,76,104,242,81,87,167,44,286,170)(50,173,275,296,154,56,52,198,301,109)(53,161,135,254,115,274,96,234,149,114)(58,97,237,256,188,128,85,240,313,277)(60,207,123,199,124,132,155,113,138,190)(66,232,233,223,218,144,130,305,222,127)(67,183,174,308,142,160,211,236,266,241)(69,279,255,73,181,157,179,156,205,146)(72,243,299,141,200,158,143,280,239,139)(103,215,212,105,108,194,206,184,197,153)(106,126,196,171,195,217,224,268,273,235)(112,213,210,246,182,122,247,282,245,147), (1,138,262,63,36,155,19,199,164,16,90,207)(2,66,80,9,45,156,54,144,165,65,15,255)(3,91,13,46,99,64,55,68,79,137,163,107)(4,224,133,173,51,245,39,126,134,52,101,246)(5,146,115,47,61,222,32,181,114,24,136,223)(6,25,120,140)(7,72,88,34,167,141,121,158,42,84,76,239)(8,206,41,197,170,37,23,215,75,105,81,26)(10,179,292,28,305,289,49,279,294,17,233,261)(11,211,20,97,33,142,94,183,43,85,83,241)(12,111,18,117,185,128,95,29,214,93,31,58)(14,122,59,118,161,235,30,112,21,77,96,195)(22,177,309,104,57,291,62,100,249,44,102,288)(27,296,234,71,313,180,110,301,135,48,256,89)(35,280,297,274,308,307,87,299,302,53,266,250)(38,194,253,265,132,298,92,103,311,178,60,290)(40,56,74,50)(67,228,152,106,151,109,160,221,70,217,258,154)(69,248,310,108,240,320,157,238,314,153,237,300)(73,260,281,113,149,270,205,202,193,123,254,269)(78,171,226,98,282,312,264,273,244,162,210,257)(82,168,284,189,187,303,172,169,252,116,209,227)(86,317,267,271,319,263,176,306,166,119,295,186)(124,198,293,232,229,272,190,275,315,130,145,316)(125,147,148,188,200,201,191,182,251,277,139,204)(127,219,131,212,286,175,218,283,230,184,242,276)(129,143,150,287,196,225,231,243,259,192,268,203)(159,304,247,208,278,174,220,318,213,216,285,236));