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 |
| prime radical | #55471 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4339 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)(11,12)(13,21)(14,15)(16,18)(17,23)(19,22)(20,24)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(2,5)(3,8)(10,15)(12,16)(17,21)(20,22)(25,27,31,35)(26,30,32,40)(28,36,37,29)(33,38,39,34);(1,3)(2,6)(4,5)(7,8)(10,15)(11,18)(17,21)(19,24)(26,32)(27,35)(28,37)(34,38);(1,4,7,6)(2,3,5,8)(9,11,14,18)(10,16,15,12)(13,19,23,24)(17,22,21,20)(25,28)(26,33)(27,36)(29,35)(30,38)(31,37)(32,39)(34,40);(1,5,7,2)(3,4,8,6)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,29,31,36)(26,34,32,38)(27,37,35,28)(30,39,40,33);(1,6,7,4)(2,8,5,3)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,30)(26,35)(27,32)(28,38)(29,33)(31,40)(34,37)(36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=g,e^2=h,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, 9034778);