Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22)(25,26,28,32,29,33,36,40)(27,30,35,37,34,31,38,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,24,19,22)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,38)(39,40);(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24);(3,8)(5,7)(10,16)(14,23)(15,19)(22,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,38)(37,40);(2,6)(5,7)(27,34)(30,31)(35,38)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=df,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315986);