Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
| direct product | C2×#7055 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C23 | #323 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
622 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 622 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4)(5,11)(6,14,15,9)(7,10)(8,12,16,18)(13,17)(19,20,21,24,22,25,27,30,23,26,28,31,29,32,33,34);(3,5,7,13)(4,8,10,16)(6,12,15,18)(9,11,14,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29419);