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 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,20)(11,21,15,17)(12,18,13,23)(14,19,24,22)(26,30)(27,33)(32,36)(34,35);(1,2)(3,7)(4,6)(5,8)(9,11,16,15)(10,17,20,21)(12,22,13,19)(14,18,24,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,40)(36,37);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,18,20,23)(11,22,15,19)(14,17,24,21)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,40)(36,37);(2,6)(5,7)(9,13,16,12)(10,18,20,23)(11,22,15,19)(14,21,24,17)(25,27,29,33)(26,31,30,38)(28,35,37,34)(32,40,36,39);(1,3)(2,7)(4,8)(5,6)(9,14)(10,19)(11,18)(12,17)(13,21)(15,23)(16,24)(20,22)(25,26)(27,31)(28,36)(29,30)(32,37)(33,38)(34,40)(35,39);(1,2)(3,5)(4,6)(7,8)(9,15,16,11)(10,17,20,21)(12,19,13,22)(14,18,24,23)(25,28,29,37)(26,32,30,36)(27,34,33,35)(31,39,38,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,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, 8596451);