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 | |
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:
(1,2,6,4)(3,7,5,8)(25,26,31,30)(27,36,39,32)(28,33,38,34)(29,35,37,40);(1,3,6,5)(2,7,4,8)(9,10,14,15)(11,16,18,23)(12,13,17,22)(19,24,21,20)(25,27)(26,32)(28,37)(29,38)(30,36)(31,39)(33,35)(34,40);(1,3,6,5)(2,8,4,7)(10,15)(11,18)(13,22)(19,21)(25,28,31,38)(26,33,30,34)(27,37,39,29)(32,35,36,40);(9,11)(10,16)(12,21)(13,20)(14,18)(15,23)(17,19)(22,24)(25,28)(26,34)(27,37)(29,39)(30,33)(31,38)(32,40)(35,36);(1,4,6,2)(3,7,5,8)(9,12,14,17)(10,13,15,22)(11,19,18,21)(16,24,23,20)(25,29,31,37)(26,35,30,40)(27,38,39,28)(32,33,36,34);(1,5,6,3)(2,7,4,8)(9,13)(10,17)(11,20)(12,15)(14,22)(16,19)(18,24)(21,23)(25,30,31,26)(27,36,39,32)(28,33,38,34)(29,40,37,35)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8826672);