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 | |
4393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(12,20)(16,21)(18,24)(22,23)(25,26,31,32)(27,29,34,30)(28,36,39,33)(35,38,40,37);(1,2,4,6)(3,8,7,5)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,32)(27,34)(33,36)(35,40);(1,3,4,7)(2,5,6,8)(10,14)(15,17)(16,21)(22,23)(25,27)(26,29)(28,35)(30,32)(31,34)(33,37)(36,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,30)(26,34,32,27)(28,37,39,38)(33,40,36,35);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,29)(26,27,32,34)(28,38,39,37)(33,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000779);