Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C42 |
| direct product | C8×#353 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,5,4,6,7,8);(9,10)(11,14,16,13)(12,15);(10,13)(11,16)(14,15)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35649);