Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C8 | C23 |
| commutator | C23 | C22×C16 |
| socle | C24 | C22×C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(10,11)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(5,6)(7,8)(9,10)(11,12);(2,4)(6,8)(9,11,12,10)Polycyclic presentation:
2^9|a^2=g,c^2=d,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55636);