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 | |
4443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,20)(11,14,19,16)(12,18,13,17)(21,24,22,23)(26,31)(28,35)(33,39)(36,37);(1,2)(3,6)(4,7)(5,8)(9,11,15,19)(10,16,20,14)(12,21,13,22)(17,24,18,23)(25,26,30,31)(27,28,32,35)(29,33,38,39)(34,36,40,37);(1,3)(2,5)(4,8)(6,7)(9,12,15,13)(10,17,20,18)(11,21,19,22)(14,23,16,24)(25,27,30,32)(26,28,31,35)(29,34,38,40)(33,36,39,37);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,28)(26,32)(27,31)(29,37)(30,35)(33,34)(36,38)(39,40);(9,13,15,12)(10,18,20,17)(11,22,19,21)(14,24,16,23);(1,2)(3,5)(4,7)(6,8)(9,14,15,16)(10,19,20,11)(12,23,13,24)(17,22,18,21)(25,29,30,38)(26,33,31,39)(27,34,32,40)(28,36,35,37)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=h,f^2=g,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, 8329237);