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 | |
4585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(9,10)(11,16)(12,14)(13,22)(15,17)(18,20)(19,23)(21,24)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(1,3,6,5)(2,7,8,4)(9,11,15,19)(10,16,17,23)(12,13,21,20)(14,22,24,18)(25,27,31,36)(26,32,35,38)(28,30,37,39)(29,33,34,40);(1,3,6,5)(2,7,8,4)(10,17)(11,19)(12,21)(18,22)(27,36)(28,37)(32,38)(33,40);(2,8)(3,5)(9,12,15,21)(10,14,17,24)(11,20,19,13)(16,18,23,22)(25,28)(26,33)(27,30)(29,38)(31,37)(32,34)(35,40)(36,39);(1,4)(2,3)(5,8)(6,7)(9,13,15,20)(10,18,17,22)(11,21,19,12)(14,23,24,16)(25,29)(26,30)(27,33)(28,38)(31,34)(32,37)(35,39)(36,40);(1,5,6,3)(2,7,8,4)(9,14)(10,12)(11,22)(13,16)(15,24)(17,21)(18,19)(20,23)(25,30)(26,34)(27,37)(28,36)(29,35)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8619902);