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 | #42986 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4470 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)(3,8,4,7)(9,10)(11,19)(12,15)(13,17)(14,22)(16,20)(18,23)(21,24)(25,26,31,29)(27,35,37,32)(28,38,39,33)(30,40,36,34);(1,3,6,4)(2,7,5,8)(9,11,15,20)(10,16,12,19)(13,23,24,14)(17,22,21,18)(25,27,31,37)(26,32,29,35)(28,36,39,30)(33,34,38,40);(10,12)(16,19)(17,21)(18,22)(25,28,31,39)(26,33,29,38)(27,30,37,36)(32,40,35,34);(1,3,6,4)(2,7,5,8)(9,12,15,10)(11,19,20,16)(13,21,24,17)(14,22,23,18)(25,28)(26,33)(27,36)(29,38)(30,37)(31,39)(32,34)(35,40);(1,4,6,3)(2,7,5,8)(9,13,15,24)(10,17,12,21)(11,14,20,23)(16,18,19,22)(25,29,31,26)(27,32,37,35)(28,38,39,33)(30,40,36,34);(1,5,6,2)(3,8,4,7)(9,14)(10,18)(11,13)(12,22)(15,23)(16,17)(19,21)(20,24)(25,30,31,36)(26,34,29,40)(27,28,37,39)(32,38,35,33)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=gi,e^2=g,f^2=ghi,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, 9619495);