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 | |
4415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,26,31,28)(27,32,37,35)(29,33,36,38)(30,34,40,39);(1,2,4,6)(3,5,8,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,34,38,39);(1,2)(3,5)(4,6)(7,8)(10,14)(11,18)(16,20)(19,24)(25,26,31,28)(27,35,37,32)(29,38,36,33)(30,34,40,39);(1,3,4,8)(2,5,6,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,28,31,26)(27,35,37,32)(29,33,36,38)(30,34,40,39);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,29,31,36)(26,33,28,38)(27,30,37,40)(32,34,35,39);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,38)(33,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575556);