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 | |
4370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,5)(3,4,8,6)(9,10,14,18)(11,15,21,23)(12,16,22,20)(13,17,24,19)(25,26,32,30)(27,34,37,31)(28,33,29,38)(35,40,36,39);(2,5)(3,8)(9,11)(10,15)(12,17)(13,20)(14,21)(16,24)(18,23)(19,22)(25,27,32,37)(26,31,30,34)(28,35,29,36)(33,39,38,40);(1,3)(2,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,15)(14,22)(17,21)(18,20)(23,24)(25,28,32,29)(26,33,30,38)(27,35,37,36)(31,39,34,40);(1,4,7,6)(2,3,5,8)(9,12)(10,16)(11,17)(13,23)(14,22)(15,24)(18,20)(19,21)(25,29,32,28)(26,33,30,38)(27,36,37,35)(31,39,34,40);(1,5,7,2)(3,4,8,6)(9,11)(10,15)(12,19)(13,16)(14,21)(17,22)(18,23)(20,24)(25,30,32,26)(27,34,37,31)(28,38,29,33)(35,40,36,39);(1,6,7,4)(2,8,5,3)(9,13)(10,17)(11,20)(12,15)(14,24)(16,21)(18,19)(22,23)(25,31,32,34)(26,27,30,37)(28,39,29,40)(33,35,38,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9121936);