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 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,7,8,4)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23);(1,3)(2,4)(5,7)(6,8)(12,20)(17,22)(19,21)(23,24)(26,31)(27,33)(32,39)(34,40);(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,26,30,31)(27,29,33,38)(28,32,36,39)(34,37,40,35);(2,5)(3,8)(9,12,15,20)(10,17,14,22)(11,19,13,21)(16,23,18,24)(25,27)(26,29)(28,34)(30,33)(31,38)(32,37)(35,39)(36,40);(1,4)(2,3)(5,8)(6,7)(9,13,15,11)(10,18,14,16)(12,21,20,19)(17,24,22,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38);(1,5,6,2)(3,7,8,4)(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,29)(26,33)(27,31)(28,35)(30,38)(32,34)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=h,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^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8602429);