Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | SD16×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | D4×#41 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,7)(3,5,8,6);(2,5)(3,8)(6,7);(9,10,12,15,13,11,16,14);(9,11)(10,13)(12,14)(15,16)98 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×GU(2,𝔽3), SD16×#444, #72342633, … |
| maximal normal subgroup of | SD16×#444, #72342633, #104×AGL(1,ℤ/8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128271);