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 | #2416 | C22 |
| 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)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26,31,35)(27,29,37,32)(28,38,40,33)(30,34,36,39);(1,3,6,4)(2,7,5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27)(26,32)(28,36)(29,35)(30,40)(31,37)(33,39)(34,38);(10,14)(15,17)(16,20)(23,24)(25,28,31,40)(26,33,35,38)(27,36,37,30)(29,34,32,39);(1,3,6,4)(2,7,5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,29,31,32)(26,27,35,37)(28,39,40,34)(30,38,36,33);(1,4,6,3)(2,7,5,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,30)(26,34)(27,28)(29,38)(31,36)(32,33)(35,39)(37,40);(1,5,6,2)(3,8,4,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,24)(19,21)(25,27)(26,32)(28,36)(29,35)(30,40)(31,37)(33,39)(34,38)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=hi,d^2=g,e^2=gh,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, 9619388);