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 | |
4394 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)(3,4)(5,8)(6,7)(9,10)(11,20)(12,23)(13,14)(15,17)(16,22)(18,21)(19,24)(25,26,31,35)(27,32,36,38)(28,30,34,40)(29,33,39,37);(1,3,6,5)(2,4,7,8)(9,11,15,22)(10,16,17,20)(12,14,21,24)(13,23,19,18)(25,27)(26,32)(28,37)(29,30)(31,36)(33,34)(35,38)(39,40);(1,4,6,8)(2,5,7,3)(10,17)(13,19)(16,20)(18,23)(27,36)(29,39)(32,38)(33,37);(1,4,6,8)(2,3,7,5)(9,12)(10,18)(11,14)(13,16)(15,21)(17,23)(19,20)(22,24)(25,28)(26,30)(27,37)(29,32)(31,34)(33,36)(35,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,19)(10,14,17,24)(11,18,22,23)(12,20,21,16)(25,29)(26,33)(27,30)(28,38)(31,39)(32,34)(35,37)(36,40);(1,5,6,3)(2,4,7,8)(9,14)(10,19)(11,21)(12,22)(13,17)(15,24)(16,18)(20,23)(25,30)(26,34)(27,29)(28,35)(31,40)(32,33)(36,39)(37,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8974719);