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 | |
4543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10,16,14)(11,18,21,15)(12,17,13,22)(19,24,20,23)(26,31)(27,32)(29,37)(35,39);(2,5)(3,8)(9,11,16,21)(10,15,14,18)(12,19,13,20)(17,23,22,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,3,6,8)(2,7,5,4)(9,12,16,13)(10,17,14,22)(11,19,21,20)(15,23,18,24)(25,27,30,32)(26,28,31,36)(29,34,37,40)(33,35,38,39);(1,4)(2,3)(5,8)(6,7)(9,13,16,12)(10,17,14,22)(11,20,21,19)(15,23,18,24)(25,26)(27,28)(29,38)(30,31)(32,36)(33,37)(34,39)(35,40);(1,5,6,2)(3,4,8,7)(9,14,16,10)(11,18,21,15)(12,22,13,17)(19,24,20,23)(25,28)(26,32)(27,31)(29,39)(30,36)(33,34)(35,37)(38,40);(1,4)(2,3)(5,8)(6,7)(9,15,16,18)(10,11,14,21)(12,23,13,24)(17,19,22,20)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,40)(36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9138090);