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 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,15)(11,12,17,16)(13,18,23,24)(19,20,21,22)(26,28)(31,34)(32,36)(38,40);(2,5)(6,7)(10,15)(11,17)(18,24)(19,21)(27,33)(31,34)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,16,15,12)(13,21,23,19)(18,20,24,22)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,24,21)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(9,13,14,23)(10,18,15,24)(11,19,17,21)(12,20,16,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=gh,e^2=gh,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, 8602409);