Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24)(25,26,30,32,29,31,28,27);(1,2,4,5)(3,8,7,6)(9,11,12,15)(10,13,14,16)(19,24)(21,23)(26,31)(27,32);(10,14)(13,16)(17,19,20,24)(18,21,22,23)(25,27,28,31,29,32,30,26);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,22)(21,23)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=efg,c^2=gh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=dei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298169);