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 | |
4620 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)(11,19)(12,17)(13,20)(14,15)(16,22)(18,24)(21,23)(25,26,31,33)(27,32,34,29)(28,30,37,40)(35,36,39,38);(1,2)(3,5)(4,6)(7,8)(10,15)(12,23)(16,19)(18,20)(25,27,31,34)(26,32,33,29)(28,35,37,39)(30,36,40,38);(1,2,4,6)(3,5,7,8)(10,15)(13,24)(16,19)(17,21)(26,33)(27,34)(30,40)(35,39);(3,7)(5,8)(9,11,14,22)(10,16,15,19)(12,18,23,20)(13,21,24,17)(25,28)(26,30)(27,35)(29,38)(31,37)(32,36)(33,40)(34,39);(1,3)(2,5)(4,7)(6,8)(9,12,14,23)(10,17,15,21)(11,20,22,18)(13,16,24,19)(25,29,31,32)(26,27,33,34)(28,36,37,38)(30,39,40,35);(1,2)(3,5)(4,6)(7,8)(9,13,14,24)(10,18,15,20)(11,21,22,17)(12,16,23,19)(25,30)(26,28)(27,36)(29,39)(31,40)(32,35)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805462);