Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C23×C43 |
| direct product | C42×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C43 | D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C43 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2538 conjugacy classes of subgroups; only the 906 normal subgroups are shown.
This group has 2538 conjugacy classes of subgroups — too many to draw. Only its 906 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) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6,7,8);(9,10,12,15,13,11,16,14);(9,11)(10,13)(12,14)(15,16)224 conjugacy classes — view the full character table.
| maximal subgroup of | C42×#857, #104×C42⋊C3, #72225944, … |
| maximal normal subgroup of | #104×C42⋊C3, #72225944, #1042, … |
| maximal quotient of | C4×#10923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260382);