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 | |
4304 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)(3,8)(4,6)(5,7)(9,10,16,19)(11,20,15,17)(12,14,13,18)(21,23,22,24)(26,31)(28,37)(32,38)(35,40);(1,3,5,6)(2,4,7,8)(9,11,16,15)(10,17,19,20)(12,22,13,21)(14,23,18,24)(27,34)(32,38)(35,40)(36,39);(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,23,20,24)(25,26,30,31)(27,32,34,38)(28,33,37,29)(35,39,40,36);(1,4,5,8)(2,3,7,6)(9,13,16,12)(10,14,19,18)(11,22,15,21)(17,24,20,23)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(1,2)(3,4)(5,7)(6,8)(9,14)(10,12)(11,23)(13,19)(15,24)(16,18)(17,22)(20,21)(25,28)(26,33)(27,35)(29,31)(30,37)(32,39)(34,40)(36,38);(1,4)(2,6)(3,7)(5,8)(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,24,18,23)(25,29,30,33)(26,28,31,37)(27,36,34,39)(32,35,38,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8570087);