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 | |
4655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23);(1,2,6,5)(3,8,4,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,26,30,31)(27,29,33,38)(28,32,37,39)(34,36,35,40);(1,3,6,4)(2,7,5,8)(10,14)(15,17)(16,20)(23,24)(26,31)(29,38)(32,39)(36,40);(1,4,6,3)(2,8,5,7)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,27,30,33)(26,29,31,38)(28,35,37,34)(32,40,39,36);(1,5,6,2)(3,8,4,7)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,39)(33,35)(38,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,29)(26,33)(27,31)(28,36)(30,38)(32,35)(34,39)(37,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202783);