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 | |
4498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,19)(12,18)(21,22)(26,31)(28,32)(33,39)(36,40);(1,2)(3,8)(4,6)(5,7)(9,10,14,15)(11,16,19,17)(12,13,18,23)(20,21,24,22)(26,31)(27,34)(33,39)(35,37);(1,3,5,6)(2,4,7,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,17)(18,24)(22,23)(25,26,30,31)(27,32,34,28)(29,33,38,39)(35,40,37,36);(1,2)(3,4)(5,7)(6,8)(9,11)(10,17)(12,20)(13,22)(14,19)(15,16)(18,24)(21,23)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,38)(39,40);(9,12,14,18)(10,13,15,23)(11,20,19,24)(16,21,17,22)(25,28,30,32)(26,27,31,34)(29,36,38,40)(33,35,39,37);(1,4)(2,6)(3,7)(5,8)(9,13)(10,18)(11,21)(12,15)(14,23)(16,24)(17,20)(19,22)(25,29,30,38)(26,33,31,39)(27,35,34,37)(28,36,32,40)Polycyclic presentation:
2^9|b^2=g,c^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8332119);