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 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,8)(3,5,6,4)(9,10)(11,18)(12,20)(13,15)(14,16)(17,22)(19,23)(21,24)(26,30)(27,33)(32,35)(34,39);(1,3,7,6)(2,4,8,5)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(26,30)(31,37)(32,35)(38,40);(1,4,7,5)(2,3,8,6)(10,16)(15,18)(17,20)(23,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(1,5,7,4)(2,3,8,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,6,7,3)(2,4,8,5)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,28,29,36)(26,32,30,35)(27,34,33,39)(31,38,37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,d^2=ghi,e^2=h,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, 8095226);