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 | |
4493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,18)(13,23)(15,17)(16,19)(20,21)(22,24)(25,26,31,35)(27,32,37,39)(28,30,38,33)(29,34,36,40);(1,2)(3,8)(4,6)(5,7)(9,11,16,15)(10,17,19,14)(12,20,13,22)(18,24,23,21)(25,27)(26,32)(28,29)(30,34)(31,37)(33,40)(35,39)(36,38);(1,3,5,6)(2,4,7,8)(9,12,16,13)(10,18,19,23)(11,20,15,22)(14,21,17,24)(25,28)(26,33)(27,36)(29,37)(30,35)(31,38)(32,34)(39,40);(1,4,5,8)(2,3,7,6)(9,13,16,12)(10,18,19,23)(11,20,15,22)(14,24,17,21)(25,29,31,36)(26,34,35,40)(27,28,37,38)(30,39,33,32);(1,2)(3,4)(5,7)(6,8)(9,14)(10,11)(12,21)(13,24)(15,19)(16,17)(18,20)(22,23)(25,30,31,33)(26,28,35,38)(27,34,37,40)(29,39,36,32);(1,4)(2,6)(3,7)(5,8)(9,15,16,11)(10,17,19,14)(12,22,13,20)(18,24,23,21)(25,28)(26,30)(27,29)(31,38)(32,34)(33,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8604770);