Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C22×#2397 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
20898 conjugacy classes of subgroups; only the 4418 normal subgroups are shown.
This group has 20898 conjugacy classes of subgroups — too many to draw. Only its 4418 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2);(3,4);(5,6,7,8)(13,14,16,17)(15,18,19,20);(6,8)(9,10)(11,12)(14,17)(18,20);(6,8)(10,12);(13,15,16,19)(14,18,17,20)176 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#10012, C2×C4×#2397, C2×#111495, … |
| maximal normal subgroup of | C22×#10012, C2×C4×#2397, C2×#111495, … |
| maximal quotient of | C22×#23798, C2×C4×#2397, C2×#6432606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534894);