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 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,15,19)(11,16,20,23)(12,17,22,21)(13,14,18,24)(25,26,31,35)(27,36,34,30)(28,39,38,32)(29,33,37,40);(1,2,5,8)(3,6,4,7)(9,11)(10,16)(12,14)(13,21)(15,20)(17,18)(19,23)(22,24)(25,27,31,34)(26,30,35,36)(28,37,38,29)(32,40,39,33);(1,3,5,4)(2,6,8,7)(11,20)(12,22)(16,23)(17,21)(25,28,31,38)(26,32,35,39)(27,37,34,29)(30,40,36,33);(9,12)(10,17)(11,14)(13,23)(15,22)(16,18)(19,21)(20,24)(25,28)(26,32)(27,37)(29,34)(30,40)(31,38)(33,36)(35,39);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,21)(12,16)(15,18)(17,20)(19,24)(22,23)(25,29,31,37)(26,33,35,40)(27,38,34,28)(30,39,36,32);(1,4,5,3)(2,7,8,6)(9,14)(10,18)(11,12)(13,19)(15,24)(16,17)(20,22)(21,23)(25,30)(26,34)(27,35)(28,40)(29,32)(31,36)(33,38)(37,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,e^2=g,f^2=gh,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, 9035849);