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 | |
4515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,8)(3,5,6,4)(25,26)(27,28)(29,37)(30,34)(31,32)(33,39)(35,40)(36,38);(1,3,7,6)(2,4,8,5)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,31,30)(26,28,32,34)(29,35,39,38)(33,36,37,40);(1,4,7,5)(2,3,8,6)(10,14)(15,17)(16,21)(22,23)(26,32)(27,30)(33,37)(35,38);(1,5,7,4)(2,3,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,31,34)(26,27,32,30)(29,36,39,40)(33,35,37,38);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,31,39)(26,33,32,37)(27,35,30,38)(28,36,34,40);(1,6,7,3)(2,4,8,5)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,30,31,27)(26,34,32,28)(29,38,39,35)(33,40,37,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,e^2=i,f^2=gh,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, 8101080);