Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12366 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#3292 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 980 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,8,10,18)(5,12)(6,15)(9,14,11,17)(13,16);(5,13)(6,15)(8,14)(9,11)(12,16)(17,18);(3,4)(5,14)(6,9)(7,10)(8,13)(11,15)(12,18)(16,17)56 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#12366, C2×#489347, #72215864, … |
| maximal normal subgroup of | C2×#489347, #72215864, #72219583, … |
| maximal quotient of | D3×#12366, #72219583, #72219624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396126);