Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22 |
| direct product | SD16×C32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | D4 |
| commutator | C4 | C22×C32 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | D4×C16 |
| prime radical | C2×D4 | C2×C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,4,5,6,7,8);(2,4)(3,7)(6,8);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)224 conjugacy classes — view the full character table.
| maximal subgroup of | C32×GL(2,𝔽3), C3×SD16×C32, C32×AΓL(1,𝔽9) |
| maximal normal subgroup of | C3×SD16×C32 |
| maximal quotient of | C3×SD16×C32, C32×AΓL(1,𝔽9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58283);