Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D64 |
| commutator | C64 | C2×C4 |
| Frattini, cosocle | C2×C64 | C22 |
| socle | C2 | #7545 |
| prime radical | C22 | Q128 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
35 conjugacy classes of subgroups.
| minimal degree | μ(G) = 256 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 256 points:
(1,2,5,13,10,17,9,48)(3,18,21,69,26,73,25,99)(4,27,29,100,33,104,32,117)(6,37,34,118,41,123,40,128)(7,42,35,120,45,131,44,132)(8,46,36,67,47,135,43,16)(11,49,50,84,54,83,136,20)(12,55,56,79,59,78,137,19)(14,62,60,108,64,107,138,28)(15,65,61,125,66,124,130,38)(22,88,85,191,92,196,91,201)(23,93,86,193,96,203,95,204)(24,97,87,170,98,206,94,72)(30,111,109,217,114,219,113,220)(31,115,110,214,116,221,112,103)(39,126,119,223,127,226,105,122)(51,145,143,235,147,238,232,172)(52,148,144,236,149,241,227,171)(53,82,133,142,150,190,230,187)(57,155,154,242,156,243,211,176)(58,77,134,153,157,182,231,179)(63,106,121,158,159,216,225,215)(68,140,161,175,164,174,207,74)(70,139,165,186,168,185,208,80)(71,160,166,198,169,197,202,89)(75,102,151,210,178,213,177,218)(76,101,152,209,181,212,180,222)(81,129,141,224,189,228,188,229)(90,199,192,251,200,253,183,195)(146,239,162,237,240,255,244,256)(163,173,205,234,245,248,254,247)(167,184,194,233,246,250,252,249);(1,3,19,74,6,22,75,171,38,89,176,239,106,184,31,82,8,24,77,173,39,90,109,141,35,86,152,145,62,139,29,49,5,21,55,140,34,85,151,148,65,160,155,237,158,233,110,142,36,87,153,234,119,192,114,189,45,96,181,235,108,186,33,84,10,26,79,175,41,92,178,236,125,198,242,255,216,250,116,190,47,98,182,248,127,200,113,188,44,95,180,238,107,185,32,83,9,25,78,174,40,91,177,241,124,197,243,256,215,249,112,187,43,94,179,247,105,183,30,81,7,23,76,172,28,80,4,20)(2,11,27,70,14,51,101,93,42,129,111,195,122,163,58,72,16,53,103,167,63,146,211,202,130,227,218,201,128,207,137,99,48,136,117,208,138,232,222,204,132,229,220,253,226,254,231,206,135,230,221,252,225,244,156,169,66,149,213,196,123,164,59,73,17,54,104,168,64,147,212,203,131,228,219,251,223,245,157,170,67,150,214,246,159,240,154,166,61,144,210,191,118,161,56,69,13,50,100,165,60,143,209,193,120,224,217,199,126,205,134,97,46,133,115,194,121,162,57,71,15,52,102,88,37,68,12,18)Polycyclic presentation:
2^9|a^2=d,b^2=ce,c^2=ef,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=hi134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2038);