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 | |
4500 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)(9,10,15,19)(11,20,22,16)(12,21,23,17)(13,14,18,24)(25,26)(27,30)(28,31)(29,33)(32,34)(35,38)(36,37)(39,40);(1,2,4,6)(3,5,8,7)(9,11,15,22)(10,16,19,20)(12,13,23,18)(14,21,24,17)(25,27,31,34)(26,32,28,30)(29,35,37,40)(33,39,36,38);(1,2)(3,5)(4,6)(7,8)(12,23)(13,18)(14,24)(17,21)(26,28)(29,37)(30,32)(35,40);(1,3,4,8)(2,5,6,7)(9,12,15,23)(10,17,19,21)(11,18,22,13)(14,16,24,20)(25,28,31,26)(27,32,34,30)(29,33,37,36)(35,38,40,39);(9,13,15,18)(10,14,19,24)(11,12,22,23)(16,17,20,21)(25,29,31,37)(26,33,28,36)(27,35,34,40)(30,38,32,39);(9,14)(10,18)(11,21)(12,16)(13,19)(15,24)(17,22)(20,23)(25,30,31,32)(26,34,28,27)(29,38,37,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8970492);