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×#7207 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C22 | #1282 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 66 points:
(1,2);(3,4,7,13,9,16,29,40,10,15,8,14,28,39,30,41)(5,17,19,42,21,45,50,64,22,44,20,43,49,63,51,65)(6,23,24,38,25,52,53,62,26,12,27,35,54,36,55,60)(11,31,32,56,34,57,58,18,33,46,37,47,59,66,61,48);(3,5,8,20,9,21,30,51,10,22,7,19,28,49,29,50)(4,11,14,32,16,34,41,58,15,33,13,37,39,59,40,61)(6,18,24,46,25,47,53,66,26,48,27,31,54,56,55,57)(12,17,35,43,36,45,60,65,23,44,38,42,52,63,62,64)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34277);