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 | |
4779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,24)(22,23);(2,6)(3,7)(10,15)(12,23)(16,19)(18,21)(25,26,30,31)(27,29,33,38)(28,32,35,39)(34,37,36,40);(1,2,5,6)(3,8,7,4)(10,15)(11,20)(12,23)(13,24)(26,31)(28,35)(29,38)(34,36);(2,6)(3,7)(9,11,14,20)(10,16,15,19)(12,18,23,21)(13,22,24,17)(25,27,30,33)(26,29,31,38)(28,36,35,34)(32,40,39,37);(1,3,5,7)(2,4,6,8)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,16,24,19)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40);(1,4)(2,7)(3,6)(5,8)(9,13,14,24)(10,18,15,21)(11,22,20,17)(12,16,23,19)(25,29)(26,33)(27,31)(28,37)(30,38)(32,36)(34,39)(35,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,d^2=hi,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=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9415804);