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 | |
4337 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)(3,7)(4,6)(5,8)(9,10,15,18)(11,19,21,16)(12,22,23,14)(13,17,24,20)(25,26)(27,32)(28,29)(30,39)(31,35)(33,38)(34,36)(37,40);(1,2,4,6)(3,5,8,7)(9,11,15,21)(10,16,18,19)(12,20,23,17)(13,22,24,14)(25,27,31,37)(26,32,35,40)(28,36,38,30)(29,34,33,39);(1,2)(3,5)(4,6)(7,8)(25,28,31,38)(26,33,35,29)(27,30,37,36)(32,34,40,39);(1,3,4,8)(2,5,6,7)(9,12)(10,14)(11,20)(13,16)(15,23)(17,21)(18,22)(19,24)(25,28)(26,29)(27,36)(30,37)(31,38)(32,34)(33,35)(39,40);(9,13,15,24)(10,17,18,20)(11,14,21,22)(12,16,23,19)(25,29)(26,28)(27,34)(30,40)(31,33)(32,36)(35,38)(37,39);(9,14)(10,12)(11,13)(15,22)(16,20)(17,19)(18,23)(21,24)(25,30,31,36)(26,34,35,39)(27,28,37,38)(29,32,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,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=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770612);