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 | #53014 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,15)(11,20,14,17)(12,19,13,18)(21,23,22,24)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(1,2)(3,8)(4,6)(5,7)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(26,31)(27,33)(32,38)(34,39);(1,3,5,6)(2,4,7,8)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(1,2)(3,4)(5,7)(6,8)(9,13,16,12)(10,19,15,18)(11,22,14,21)(17,24,20,23)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(9,14,16,11)(10,20,15,17)(12,22,13,21)(18,24,19,23);(1,4)(2,6)(3,7)(5,8)(9,15,16,10)(11,20,14,17)(12,19,13,18)(21,24,22,23)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8331829);