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 | |
4638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(6,7)(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(25,26)(27,32)(28,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,2,5,3)(4,6,8,7)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(25,27,31,30)(26,32,37,36)(28,35,39,38)(29,34,40,33);(1,2,5,3)(4,6,8,7)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,28)(26,33)(27,38)(29,36)(30,35)(31,39)(32,40)(34,37);(1,2,5,3)(4,7,8,6)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,28)(26,34)(27,35)(29,36)(30,38)(31,39)(32,40)(33,37);(1,3,5,2)(4,6,8,7)(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23)(25,29)(26,35)(27,34)(28,36)(30,33)(31,40)(32,39)(37,38);(1,4,5,8)(2,6,3,7)(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,30,31,27)(26,36,37,32)(28,35,39,38)(29,34,40,33)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,e^2=g,f^2=h,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, 8599039);