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 | |
4431 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)(9,10,15,19)(11,16,14,18)(12,13,17,23)(20,21,22,24)(26,31)(27,35)(28,38)(34,39);(3,8)(6,7)(9,11,15,14)(10,16,19,18)(12,20,17,22)(13,21,23,24)(25,26)(27,32)(28,33)(29,39)(30,31)(34,36)(35,37)(38,40);(1,3,5,8)(2,6,4,7)(11,14)(12,17)(13,23)(16,18)(25,27,30,35)(26,32,31,37)(28,29,38,36)(33,39,40,34);(3,8)(6,7)(9,12)(10,13)(11,20)(14,22)(15,17)(16,21)(18,24)(19,23)(25,28,30,38)(26,33,31,40)(27,36,35,29)(32,34,37,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,17)(11,21)(12,19)(14,24)(15,23)(16,22)(18,20)(25,29,30,36)(26,34,31,39)(27,28,35,38)(32,40,37,33);(1,4,5,2)(3,6,8,7)(9,14,15,11)(10,18,19,16)(12,22,17,20)(13,24,23,21)(25,26)(27,37)(28,33)(29,34)(30,31)(32,35)(36,39)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8930835);