Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13681 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | C4⋊Q8 | C24 |
| prime radical | C2×#3291 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1066 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 114 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);(3,4,9,8)(5,13,16,18)(6,17,12,7)(10,15,11,14);(3,5)(4,10)(6,15)(7,18)(8,11)(9,16)(12,14)(13,17);(3,5)(4,11)(6,13)(7,14)(8,10)(9,16)(12,18)(15,17)50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153628, C2×#153642, C2×#512900, … |
| maximal normal subgroup of | C2×#153628, C2×#153642, C2×#512900, … |
| maximal quotient of | #72283624 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420086);