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 | C23×#399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C23 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C27 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
25274 conjugacy classes of subgroups; only the 6658 normal subgroups are shown.
This group has 25274 conjugacy classes of subgroups — too many to draw. Only its 6658 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 — 6 generators on 22 points:
(1,2);(3,4);(5,6);(11,12,13,14)(17,21)(19,22);(8,10)(12,14)(15,16,18,20)(17,22,21,19);(7,8)(9,10)(11,12)(13,14)(15,17)(16,19)(18,21)(20,22)176 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#2668, C2×#156769, C2×#158359, … |
| maximal normal subgroup of | C23×#2668, C2×#156769, C2×#158359, … |
| maximal quotient of | C2×D4×#399, C23×#1439, C23×#1444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534890);