Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7191 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C23 | #323 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(7,8,9,13,10,14,16,20,11,12,15,18,17,19,21,22);(3,4,5,6)(8,12)(14,19)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34115);