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 | |
4594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(25,26)(27,36)(28,38)(29,37)(30,33)(31,32)(34,39)(35,40);(2,5)(6,7)(9,10)(11,14)(12,15)(13,16)(17,22)(18,23)(19,21)(20,24)(26,32)(28,39)(33,36)(35,37);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,23)(12,15,21,19)(17,22,20,24)(25,27,31,30)(26,33,32,36)(28,35,39,37)(29,34,40,38);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22)(25,28,31,39)(26,34,32,38)(27,37,30,35)(29,36,40,33);(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,24,19,22)(25,29)(26,35)(27,34)(28,36)(30,38)(31,40)(32,37)(33,39);(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24)(25,30,31,27)(26,33,32,36)(28,37,39,35)(29,34,40,38)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8633706);