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 | |
4614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,13)(14,19)(15,17)(18,23)(20,24)(21,22)(25,26)(27,37)(28,34)(29,31)(30,35)(32,36)(33,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,15,20)(10,16,17,24)(12,21,18,19)(13,22,23,14)(25,27,32,40)(26,33,36,37)(28,38,29,35)(30,31,39,34);(2,3)(6,7)(10,17)(11,20)(13,23)(19,21)(25,28,32,29)(26,34,36,31)(27,38,40,35)(30,37,39,33);(4,8)(6,7)(9,12,15,18)(10,13,17,23)(11,19,20,21)(14,24,22,16)(25,29,32,28)(26,34,36,31)(27,35,40,38)(30,37,39,33);(1,3,5,2)(4,7,8,6)(9,13)(10,18)(11,14)(12,17)(15,23)(16,21)(19,24)(20,22)(25,30,32,39)(26,35,36,38)(27,34,40,31)(28,33,29,37);(1,4)(2,6)(3,7)(5,8)(9,14)(10,19)(11,13)(12,16)(15,22)(17,21)(18,24)(20,23)(25,31,32,34)(26,29,36,28)(27,39,40,30)(33,35,37,38)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520366);