Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94835 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7537 |
| commutator | C64 | C2×C4 |
| Frattini, cosocle | C2×C64 | C22 |
| prime radical | C22 | SD128 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
51 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 256 points:
(1,2,5,13,10,17,9,53)(3,18,20,80,25,84,24,104)(4,26,29,105,34,109,33,134)(6,38,35,135,44,62,43,151)(7,45,36,137,49,153,48,157)(8,50,37,78,52,159,47,16)(11,54,55,19,59,86,67,90)(12,60,63,125,66,124,161,30)(14,70,68,28,73,116,162,41)(15,74,69,99,76,146,152,40)(21,94,91,217,97,178,96,223)(22,98,92,218,101,177,100,226)(23,102,93,208,103,228,51,83)(27,71,110,196,87,79,114,193)(31,126,119,240,130,245,129,248)(32,131,120,233,133,249,128,108)(39,81,142,183,85,61,136,176)(42,148,117,182,150,252,147,140)(46,82,155,205,156,207,138,224)(56,166,164,141,168,236,190,112)(57,169,165,154,170,235,191,111)(58,171,77,160,172,89,192,212)(64,185,184,238,187,255,231,214)(65,123,158,175,189,243,253,216)(72,145,139,195,200,118,181,239)(75,163,201,88,202,211,194,215)(95,221,209,186,222,256,143,220)(106,174,229,115,232,237,250,122)(107,173,230,225,180,242,244,121)(113,199,234,204,213,227,132,198)(127,197,246,254,247,203,241,167)(144,219,251,179,210,188,149,206);(1,3,10,25)(2,11,17,59)(4,27,34,87)(5,24,9,20)(6,39,44,85)(7,46,49,156)(8,51,52,93)(12,61,66,81)(13,67,53,55)(14,71,73,79)(15,75,76,202)(16,77,78,192)(18,26,84,109)(19,30,90,125)(21,28,97,41)(22,99,101,40)(23,37,103,47)(29,114,33,110)(31,127,130,247)(32,132,133,234)(35,136,43,142)(36,138,48,155)(38,56,62,168)(42,149,150,251)(45,57,153,170)(50,58,159,172)(54,60,86,124)(63,183,161,176)(64,186,187,220)(65,188,189,219)(68,193,162,196)(69,194,152,201)(70,96,116,91)(72,199,200,227)(74,100,146,92)(80,134,104,105)(82,126,207,245)(83,131,208,249)(88,214,215,238)(89,216,171,175)(94,106,178,232)(95,195,222,239)(98,107,177,180)(102,108,228,233)(111,115,154,122)(112,225,141,121)(113,120,213,128)(117,210,147,144)(118,209,145,143)(119,241,129,246)(123,212,243,160)(135,190,151,164)(137,191,157,165)(139,198,181,204)(140,197,182,203)(148,167,252,254)(158,179,253,206)(163,185,211,255)(166,173,236,242)(169,174,235,237)(184,221,231,256)(205,248,224,240)(217,250,223,229)(218,244,226,230)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=efi,d^2=i,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cei,d^b=di,e^b=ef,f^b=fg,g^b=gh,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2033);