Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,14)(11,17)(16,21)(18,23)(25,26)(27,29)(28,31)(30,38)(32,34)(33,40)(35,36)(37,39);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,32,31)(26,29,34,28)(30,35,40,39)(33,37,38,36);(2,4)(6,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,32,29)(26,27,34,31)(30,36,40,37)(33,35,38,39);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,29,32,28)(26,27,34,31)(30,37,40,36)(33,35,38,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,30,32,40)(26,33,34,38)(27,35,31,39)(28,36,29,37);(1,4,5,2)(3,7,8,6)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,31,32,27)(26,28,34,29)(30,39,40,35)(33,36,38,37)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8092040);