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 | |
4484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,14)(15,20)(16,21)(22,24)(25,26)(27,38)(28,31)(29,36)(30,35)(32,37)(33,40)(34,39);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,27)(26,33)(28,29)(30,39)(31,35)(32,40)(34,36)(37,38);(1,2,4,6)(3,5,8,7)(12,19)(16,21)(17,23)(22,24)(25,28,32,39)(26,34,37,31)(27,29,40,30)(33,36,38,35);(9,10)(11,15)(12,16)(13,14)(17,22)(18,20)(19,21)(23,24)(25,29,32,30)(26,35,37,36)(27,28,40,39)(31,38,34,33);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,30,32,29)(26,36,37,35)(27,39,40,28)(31,33,34,38);(1,2)(3,5)(4,6)(7,8)(9,12,13,19)(10,16,14,21)(11,17,18,23)(15,22,20,24)(25,31,32,34)(26,28,37,39)(27,35,40,36)(29,38,30,33)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095356);