Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94829 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7530 |
| commutator | C82 | C2×C4 |
| Frattini, cosocle | #1709 | C22 |
| prime radical | #493 | D4 |
| LCS | G ▷ C82 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ #1709 ◁ G | |
| LECS | G ▷ #1709 ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,14,16,43)(4,20,23,56)(5,27,29,10)(6,12,22,11)(7,13)(8,15,32,48)(9,24)(17,53,31,45)(18,46,55,64)(19,47,28,44)(21,59,52,35)(25,57,26,58)(30,41,54,42)(33,49,36,51)(34,50,40,63)(37,61,38,60)(39,62);(1,3,7,19)(2,8)(4,21,26,50)(5,28,30,16)(6,31,22,18)(9,14)(10,40)(11,35)(12,34)(13,36)(15,25,51,23)(17,54,55,29)(20,53)(24,49,62,48)(27,33)(32,41)(37,46)(38,45)(39,47)(42,59)(43,57)(44,56)(52,60,63,61)(58,64)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fgi,e^2=gh,f^2=hi,g^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cfgh,d^b=defi,e^b=eghi,f^b=fhi,g^b=gi,d^c=dfg,e^d=eghi,f^d=fhi,g^d=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2025);