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 | |
4335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,3)(6,7)(9,10,14,15)(11,19,20,16)(12,17,23,22)(13,18,24,21)(25,26,31,35)(27,32,36,38)(28,33,37,40)(29,30,34,39);(1,2,5,3)(4,6,8,7)(10,15)(11,20)(17,22)(18,21)(27,36)(28,37)(32,38)(33,40);(1,2,5,3)(4,6,8,7)(25,27,31,36)(26,32,35,38)(28,30,37,39)(29,33,34,40);(1,2,5,3)(4,7,8,6)(9,11,14,20)(10,16,15,19)(12,21,23,18)(13,17,24,22)(25,28)(26,33)(27,30)(29,38)(31,37)(32,34)(35,40)(36,39);(1,3,5,2)(4,6,8,7)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,29)(26,30)(27,33)(28,38)(31,34)(32,37)(35,39)(36,40);(1,4,5,8)(2,6,3,7)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,30)(26,34)(27,37)(28,36)(29,35)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=hi,d^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=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, 8603423);