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:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,23)(20,22)(21,24)(25,26,29,30,28,31,27,32);(1,3,5,4)(2,6,7,8)(17,19)(18,21)(20,24)(22,23);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,14,15,16)(18,22)(19,24)(25,27,28,29)(26,30,31,32);(2,7)(6,8)(9,12,13,11)(10,14,15,16)(18,22)(21,23)(25,26,27,32,28,31,29,30)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=ef,d^2=fgh,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286908);