Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(5,6,7,8)(9,10,13,16)(11,14,18,22)(12,19,20,15)(17,23,24,21)(26,30)(31,37)(32,35)(38,39);(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(5,6)(7,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(9,12,13,20)(10,15,16,19)(11,17,18,24)(14,21,22,23)(25,28,29,36)(26,32,30,35)(27,33,34,40)(31,38,37,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327651);