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 | |
4442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10)(11,16)(12,18)(13,19)(14,24)(15,17)(20,23)(21,22)(25,26,31,34)(27,35,37,32)(28,38,39,29)(30,33,36,40);(1,2,5,4)(3,7,8,6)(9,11,15,14)(10,16,17,24)(12,20,22,19)(13,18,23,21)(25,27,31,37)(26,32,34,35)(28,33,39,40)(29,36,38,30);(1,3,5,8)(2,6,4,7)(10,17)(11,14)(12,22)(13,23)(28,39)(29,38)(30,36)(33,40);(1,4,5,2)(3,7,8,6)(9,12)(10,18)(11,20)(13,24)(14,19)(15,22)(16,23)(17,21)(25,28,31,39)(26,29,34,38)(27,33,37,40)(30,32,36,35);(9,13,15,23)(10,19,17,20)(11,21,14,18)(12,16,22,24)(25,29)(26,28)(27,36)(30,37)(31,38)(32,33)(34,39)(35,40);(9,14,15,11)(10,16,17,24)(12,19,22,20)(13,18,23,21)(25,30,31,36)(26,33,34,40)(27,29,37,38)(28,32,39,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8934505);