Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13561 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13561 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1716 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 194 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);(3,4)(5,12,14,8)(6,10,16,9)(7,11)(13,18,15,17);(3,5)(4,8)(6,15)(7,13)(9,12)(10,18)(11,17)(14,16);(5,13)(8,17)(12,18)(14,15)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#152018, C2×#152717, C2×#152679, … |
| maximal normal subgroup of | C2×#152018, C2×#152717, C2×#152679, … |
| maximal quotient of | C2×#153461, C2×#153467, #72279279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419758);