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 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,8,5,4)(9,10,14,15)(11,13,18,16)(12,20,22,17)(19,23,24,21)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(1,3,6,5)(2,4,7,8)(10,15)(11,18)(17,20)(19,24)(27,36)(28,37)(32,38)(33,40);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,27,31,36)(26,32,35,38)(28,30,37,39)(29,33,34,40);(2,7)(3,5)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21)(25,28)(26,33)(27,30)(29,38)(31,37)(32,34)(35,40)(36,39);(1,4,6,8)(2,3,7,5)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,29)(26,30)(27,33)(28,38)(31,34)(32,37)(35,39)(36,40);(1,5,6,3)(2,4,7,8)(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,30)(26,34)(27,37)(28,36)(29,35)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=h,e^2=g,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, 8603137);