Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13569 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#13574 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C4⋊Q8 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1503 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 1503 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);(3,4,7,12,9,5,14,16)(6,10,17,15,8,13,18,11);(3,5)(4,9)(6,10)(7,11)(8,13)(12,18)(14,15)(16,17);(3,6,9,8)(4,10,5,13)(7,17,14,18)(11,12,15,16)56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280274, C2×#512900, C2×#153148, … |
| maximal normal subgroup of | #72280274, C2×#512900, C2×#153148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419780);