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 | |
4273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,20)(13,14)(15,18)(16,22)(19,23)(21,24)(25,26,31,30)(27,34,29,32)(28,33,38,37)(35,40,36,39);(1,2)(3,4)(5,6)(7,8)(10,14)(15,17)(16,20)(23,24)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(2,6)(3,7)(10,14)(11,18)(16,20)(19,21)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(3,7)(4,8)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,29,31,27)(26,34,30,32)(28,36,38,35)(33,40,37,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,26,31,30)(27,32,29,34)(28,33,38,37)(35,39,36,40);(1,4)(2,3)(5,8)(6,7)(9,12,13,22)(10,16,14,20)(11,19,18,21)(15,23,17,24)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,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=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9532669);