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×#395 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C26 | C23 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
27482 conjugacy classes of subgroups; only the 7170 normal subgroups are shown.
This group has 27482 conjugacy classes of subgroups — too many to draw. Only its 7170 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) = 512 |
A minimal faithful permutation representation — 6 generators on 22 points:
(1,2);(3,4);(5,6);(12,14)(15,16,19,18)(17,21,22,20);(8,10)(12,14)(15,17,19,22)(16,20,18,21);(7,8)(9,10)(11,12)(13,14)(15,18,19,16)(17,20,22,21)176 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#2388, C23×#2430, C23×#2555, … |
| maximal normal subgroup of | C23×#2388, C23×#2430, C23×#2555, … |
| maximal quotient of | C23×#1444, C24×#395, C22×#14758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534884);