Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#10304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2768 conjugacy classes of subgroups; only the 410 normal subgroups are shown.
This group has 2768 conjugacy classes of subgroups — too many to draw. Only its 410 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(8,12)(9,10)(13,14)(15,16)(17,20)(18,21)(19,22);(4,6)(7,8)(9,13)(10,12)(11,14)(16,20)(17,18)(21,22);(8,12)(9,10)(13,14)(16,21)(17,18)(20,22)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10304⋊C3, C22×#10304, #72211642, … |
| maximal normal subgroup of | C2×#10304⋊C3, C22×#10304, #72211642, … |
| maximal quotient of | C2×#102857, C22×#10304, C2×#111526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140541);