Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C8 | C23 |
| commutator | C22 | C2×C4×C16 |
| socle, prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,8)(9,10,11,12)(13,14,16,21,18,15,20,24,19,23,17,22,27,26,28,25);(2,4)(13,15,17,25,18,23,28,21,19,26,16,24,27,14,20,22);(5,6,7,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54157);