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) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4×C8 | C23 |
| commutator | C22 | C2×C4×C16 |
| socle, prime radical | C23 | C2×C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
352 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)(5,6,8,7,9,10,12,11);(2,4)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(5,7,8,10,9,11,12,6)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36261);