Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29751 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#389 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,15,20,13,19)(11,16,24,32,22,31,23,33)(12,25,28,35,29,36,27,17)(21,34,26,39,37,40,38,30);(2,5)(6,8)(9,11,13,23,15,22,14,24)(10,16,19,33,20,31,18,32)(12,26,28,37,29,38,27,21)(17,34,25,39,35,40,36,30);(1,3)(2,6)(4,7)(5,8)(9,12,13,27,15,29,14,28)(10,17,18,25,20,35,19,36)(11,21,23,38,22,37,24,26)(16,30,32,34,31,39,33,40);(3,7)(6,8)(11,22)(16,31)(21,37)(23,24)(26,38)(30,39)(32,33)(34,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287998);