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 | |
4599 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)(6,8)(9,10,13,14)(11,15,18,22)(12,16,21,19)(17,23,20,24)(25,26)(27,29)(28,36)(30,31)(32,34)(33,38)(35,39)(37,40);(3,7)(6,8)(10,14)(15,22)(16,19)(23,24)(26,31)(27,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(25,26,30,31)(27,32,34,29)(28,33,38,36)(35,39,40,37);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,12,13,21)(10,16,14,19)(11,17,18,20)(15,23,22,24)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=gi,c^2=i,e^2=g,f^2=g,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, 8602559);