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 |
| prime radical | #50695 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23)(25,26,32,29)(27,31,34,36)(28,33,39,30)(35,37,38,40);(1,3,6,4)(2,7,5,8)(10,15)(12,23)(16,19)(18,21)(25,27)(26,31)(28,38)(29,36)(30,37)(32,34)(33,40)(35,39);(10,15)(11,20)(17,22)(18,21)(25,28,32,39)(26,33,29,30)(27,35,34,38)(31,37,36,40);(1,3,6,4)(2,7,5,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,29,32,26)(27,36,34,31)(28,30,39,33)(35,40,38,37);(1,4,6,3)(2,7,5,8)(9,12)(10,17)(11,21)(13,19)(14,23)(15,22)(16,24)(18,20)(25,30)(26,28)(27,37)(29,39)(31,38)(32,33)(34,40)(35,36);(1,5,6,2)(3,8,4,7)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,31,32,36)(26,34,29,27)(28,37,39,40)(30,35,33,38)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=i,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9619486);