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×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22)(25,26,31,28,30,33,29,39)(27,34,36,35,37,32,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,24,19,22)(25,27)(26,32)(28,38)(29,40)(30,37)(31,36)(33,34)(35,39);(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24);(2,5)(6,8)(10,16)(14,23)(15,19)(22,24)(25,28,31,33,30,39,29,26)(27,35,36,32,37,38,40,34);(1,3,4,7)(2,6,5,8)(25,29,30,31)(26,28,33,39)(27,36,37,40)(32,35,34,38)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=f,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, 6332108);