Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3314 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)(6,8)(9,10,13,14)(11,15,18,21)(12,16,20,22)(17,23,19,24)(25,26,29,33)(27,31,35,30)(28,36,38,32)(34,39,37,40);(10,14)(15,21)(16,22)(23,24)(25,27,29,35)(26,30,33,31)(28,34,38,37)(32,39,36,40);(5,6)(7,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,27)(26,31)(28,37)(29,35)(30,33)(32,39)(34,38)(36,40);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26)(27,31)(28,32)(29,33)(30,35)(34,40)(36,38)(37,39);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,28,29,38)(26,32,33,36)(27,34,35,37)(30,39,31,40)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4287684);