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 | |
4573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,17)(12,20)(13,15)(14,19)(16,21)(18,22)(23,24)(25,26,31,33)(27,34,28,30)(29,37,39,32)(35,38,36,40);(3,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,26,31,33)(27,34,28,30)(29,32,39,37)(35,40,36,38);(1,2,5,4)(3,6,7,8)(10,15)(14,17)(16,20)(22,23)(25,27,31,28)(26,30,33,34)(29,35,39,36)(32,38,37,40);(1,3,5,7)(2,6,4,8)(9,12,13,21)(10,16,15,20)(11,18,19,24)(14,22,17,23)(25,28,31,27)(26,30,33,34)(29,36,39,35)(32,38,37,40);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,29,31,39)(26,32,33,37)(27,35,28,36)(30,38,34,40);(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,21)(23,24)(25,30)(26,27)(28,33)(29,38)(31,34)(32,35)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8090994);