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 | |
4543 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,17)(13,18)(15,16)(19,24)(20,22)(21,23)(25,26,31,35)(27,32,36,39)(28,33,38,37)(29,30,34,40);(2,7)(3,8)(9,11,15,19)(10,14,16,24)(12,20,21,13)(17,22,23,18)(25,27)(26,32)(28,30)(29,37)(31,36)(33,34)(35,39)(38,40);(1,3,6,8)(2,4,7,5)(10,16)(11,19)(12,21)(18,22)(27,36)(28,38)(32,39)(33,37);(1,4,6,5)(2,3,7,8)(9,12)(10,17)(11,20)(13,19)(14,22)(15,21)(16,23)(18,24)(25,28)(26,33)(27,30)(29,39)(31,38)(32,34)(35,37)(36,40);(1,2)(3,4)(5,8)(6,7)(9,13,15,20)(10,18,16,22)(11,21,19,12)(14,23,24,17)(25,29)(26,30)(27,37)(28,32)(31,34)(33,36)(35,40)(38,39);(1,5,6,4)(2,8,7,3)(9,14)(10,19)(11,16)(12,22)(13,17)(15,24)(18,21)(20,23)(25,30)(26,34)(27,28)(29,35)(31,40)(32,33)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9034796);