Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
| direct product | C22×#1390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #99 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2);(3,4);(9,10)(11,15,16,13)(12,14);(5,6,7,8)(10,13)(14,15)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#494600, C2×#498931, C2×#146654, … |
| maximal normal subgroup of | C2×#494600, C2×#498931, C2×#146654, … |
| maximal quotient of | C2×#128073, #72236734, #72240339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400078);