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 | |
4287 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)(3,5)(4,6)(7,8)(10,14)(12,20)(15,17)(18,24)(27,33)(31,37)(34,39)(38,40);(1,2,4,6)(3,5,8,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,30)(31,37)(32,35)(38,40);(9,10,13,14)(11,15,19,17)(12,16,20,21)(18,22,24,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3,4,8)(2,5,6,7)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)Polycyclic presentation:
2^9|b^2=gh,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818196);