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 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10,13,14)(11,15,18,22)(12,16,21,19)(17,23,20,24);(1,3,6,5)(2,7,4,8)(10,14)(15,22)(16,19)(23,24)(25,26)(27,34)(28,36)(29,31)(30,33)(32,37)(35,39)(38,40);(1,3,6,5)(2,8,4,7)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,39,36,40);(1,4,6,2)(3,7,5,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,28)(26,32)(27,35)(29,38)(30,37)(31,39)(33,36)(34,40);(1,5,6,3)(2,7,4,8)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,37)(38,40);(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24)(25,29,30,27)(26,31,33,34)(28,35,37,38)(32,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607665);