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 | |
4366 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)(11,18,12,16)(13,19,20,14)(21,24,23,22)(25,26,31,33)(27,32,34,29)(28,30,37,40)(35,36,39,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,17,18)(13,23,20,21)(14,22,19,24)(25,27,31,34)(26,32,33,29)(28,35,37,39)(30,36,40,38);(1,2,4,6)(3,5,7,8)(10,17)(14,19)(16,18)(22,24)(26,33)(27,34)(30,40)(35,39);(3,7)(5,8)(9,12,15,11)(10,18,17,16)(13,21,20,23)(14,24,19,22)(25,28)(26,30)(27,35)(29,38)(31,37)(32,36)(33,40)(34,39);(1,3)(2,5)(4,7)(6,8)(9,13,15,20)(10,19,17,14)(11,21,12,23)(16,22,18,24)(25,29,31,32)(26,27,33,34)(28,36,37,38)(30,39,40,35);(1,2)(3,5)(4,6)(7,8)(9,14,15,19)(10,20,17,13)(11,22,12,24)(16,21,18,23)(25,30)(26,28)(27,36)(29,39)(31,40)(32,35)(33,37)(34,38)Polycyclic presentation:
2^9|a^2=h,b^2=h,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, 8805459);