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 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,18)(11,14,17,16)(12,21,22,13)(19,20,24,23)(25,26,31,28)(27,34,30,32)(29,33,37,36)(35,40,38,39);(1,2)(3,4)(5,7)(6,8)(9,11,15,17)(10,16,18,14)(12,19,22,24)(13,20,21,23)(29,37)(33,36)(35,38)(39,40);(1,3)(2,6)(4,7)(5,8)(25,27,31,30)(26,32,28,34)(29,35,37,38)(33,39,36,40);(1,4,5,6)(2,3,7,8)(9,12,15,22)(10,13,18,21)(11,19,17,24)(14,23,16,20)(25,28,31,26)(27,34,30,32)(29,36,37,33)(35,40,38,39);(1,3)(2,4)(5,8)(6,7)(9,13)(10,12)(11,20)(14,24)(15,21)(16,19)(17,23)(18,22)(25,29,31,37)(26,33,28,36)(27,35,30,38)(32,39,34,40);(1,2)(3,6)(4,8)(5,7)(9,14,15,16)(10,17,18,11)(12,20,22,23)(13,19,21,24)(25,30,31,27)(26,34,28,32)(29,38,37,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=ghi,e^2=h,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, 9131425);