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 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,17)(11,16,20,23)(12,14,19,24)(13,22,21,18)(25,26,31,33)(27,35,30,32)(28,38,34,29)(36,40,39,37);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,13)(14,22)(15,20)(17,23)(18,24)(19,21)(25,27,31,30)(26,32,33,35)(28,39,34,36)(29,37,38,40);(1,3)(2,5)(4,8)(6,7)(10,17)(11,20)(12,19)(18,22)(26,33)(28,34)(32,35)(36,39);(3,8)(5,6)(9,12)(10,14)(11,21)(13,20)(15,19)(16,18)(17,24)(22,23)(25,28,31,34)(26,29,33,38)(27,36,30,39)(32,40,35,37);(1,2)(3,5)(4,7)(6,8)(9,13,15,21)(10,18,17,22)(11,12,20,19)(14,16,24,23)(25,29)(26,34)(27,37)(28,33)(30,40)(31,38)(32,36)(35,39);(1,3)(2,6)(4,8)(5,7)(9,14)(10,19)(11,22)(12,17)(13,23)(15,24)(16,21)(18,20)(25,30,31,27)(26,32,33,35)(28,36,34,39)(29,37,38,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=gh,f^2=g,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, 8610028);