Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31851 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #25447 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1757 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,5,14,8,13)(3,15,18,36,21,9,20,34)(4,22,24,40,23,10,27,38)(7,12,28,41,26,43,32,30)(16,46,35,29,48,19,45,44)(17,37,49,62,51,33,52,60)(25,42,54,59,31,39,57,56)(47,53,61,58,64,50,63,55);(1,3,6,20,5,21,8,18)(2,9,13,34,14,15,11,36)(4,17,24,52,23,51,27,49)(7,29,28,46,26,44,32,19)(10,37,40,62,22,33,38,60)(12,16,30,35,43,48,41,45)(25,55,54,50,31,58,57,53)(39,47,59,61,42,64,56,63);(1,2)(3,16)(4,10)(5,14)(6,13)(7,30)(8,11)(9,29)(12,32)(15,44)(17,47)(18,45)(19,36)(20,35)(21,48)(22,23)(24,40)(25,56)(26,41)(27,38)(28,43)(31,59)(33,58)(34,46)(37,55)(39,54)(42,57)(49,63)(50,62)(51,64)(52,61)(53,60);(1,4,6,27,5,23,8,24)(2,10,13,38,14,22,11,40)(3,17,20,49,21,51,18,52)(7,31,28,54,26,25,32,57)(9,33,34,62,15,37,36,60)(12,42,30,59,43,39,41,56)(16,47,35,63,48,64,45,61)(19,53,29,58,46,50,44,55)Polycyclic presentation:
2^9|a^2=f,b^2=efi,d^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,f^d=fi,g^d=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417902);