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#153612

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monolithic
• monomial
• stem
• ambivalent

Aliases

No aliases for this group.

Subgroups

171 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 256 points:

(1,2,10,18)(3,19,25,81)(4,26,31,68)(5,32,39,121)(6,17,9,14)(7,43,47,144)(8,48,50,70)(11,52,57,21)(12,40,61,28)(13,62,67,36)(15,71,74,34)(16,42,76,45)(20,82,88,189)(22,91,51,79)(23,93,94,207)(24,95,96,204)(27,101,107,215)(29,110,112,230)(30,113,115,60)(33,72,125,185)(35,120,38,117)(37,134,136,234)(41,140,46,143)(44,63,146,173)(49,80,148,191)(53,152,156,195)(54,89,77,149)(55,158,159,124)(56,160,161,123)(58,162,165,223)(59,166,168,222)(64,130,66,133)(65,176,178,203)(69,127,73,128)(75,151,187,90)(78,87,190,84)(83,97,150,155)(85,199,201,247)(86,188,202,252)(92,205,145,177)(98,170,174,105)(99,183,180,103)(100,139,181,109)(102,192,220,248)(104,116,129,212)(106,225,227,233)(108,142,138,229)(111,193,232,245)(114,194,198,208)(118,171,175,126)(119,131,235,132)(122,186,137,182)(135,184,237,157)(141,179,147,172)(153,226,240,256)(154,238,241,196)(163,231,246,228)(164,221,213,250)(167,219,249,216)(169,217,210,243)(197,206,200,254)(209,242,239,218)(211,244,251,253)(214,236,255,224);(1,3)(2,11)(4,20)(5,33)(6,22)(7,44)(8,49)(9,51)(10,25)(12,53)(13,63)(14,54)(15,72)(16,75)(17,77)(18,57)(19,32)(21,36)(23,34)(24,45)(26,97)(27,102)(28,84)(29,111)(30,114)(31,88)(35,122)(37,135)(38,137)(39,125)(40,87)(41,141)(42,96)(43,55)(46,147)(47,146)(48,56)(50,148)(52,62)(58,163)(59,167)(60,169)(61,156)(64,172)(65,177)(66,179)(67,173)(68,155)(69,182)(70,161)(71,94)(73,186)(74,185)(76,187)(78,116)(79,117)(80,134)(81,121)(82,101)(83,105)(85,103)(86,109)(89,130)(90,203)(91,120)(92,127)(93,118)(95,119)(98,193)(99,192)(100,154)(104,216)(106,226)(107,220)(108,228)(110,209)(112,232)(113,210)(115,198)(123,126)(124,132)(128,145)(129,219)(131,158)(133,149)(136,237)(138,231)(139,202)(140,184)(142,153)(143,157)(144,159)(150,170)(151,176)(152,162)(160,171)(164,247)(165,246)(166,200)(168,249)(174,245)(175,207)(178,205)(180,248)(181,241)(183,201)(188,214)(189,215)(190,212)(191,234)(194,225)(195,223)(196,253)(197,222)(199,213)(204,235)(206,211)(208,233)(217,221)(218,224)(227,256)(229,240)(230,239)(236,242)(238,244)(243,250)(251,254)(252,255);(1,4,10,31)(2,12,18,61)(3,20,25,88)(5,27,39,107)(6,40,9,28)(7,29,47,112)(8,30,50,115)(11,53,57,156)(13,58,67,165)(14,68,17,26)(15,59,74,168)(16,60,76,113)(19,78,81,190)(21,83,52,150)(22,87,51,84)(23,85,94,201)(24,86,96,202)(32,116,121,212)(33,102,125,220)(34,103,71,183)(35,129,38,104)(36,105,62,170)(37,106,136,227)(41,138,46,108)(42,139,45,109)(43,142,144,229)(44,111,146,232)(48,100,70,181)(49,114,148,198)(54,155,77,97)(55,153,159,240)(56,154,161,241)(63,163,173,246)(64,174,66,98)(65,164,178,213)(69,180,73,99)(72,167,185,249)(75,169,187,210)(79,189,91,82)(80,188,191,252)(89,152,149,195)(90,196,151,238)(92,200,145,197)(93,206,207,254)(95,208,204,194)(101,117,215,120)(110,143,230,140)(118,211,175,251)(119,233,235,225)(122,219,137,216)(123,217,160,243)(124,218,158,242)(126,221,171,250)(127,166,128,222)(130,162,133,223)(131,236,132,224)(134,214,234,255)(135,226,237,256)(141,231,147,228)(157,239,184,209)(172,245,179,193)(176,244,203,253)(177,247,205,199)(182,248,186,192)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fg,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi,e^c=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60808);