Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211×C4 |
| direct product | C2×#13851 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C25 | C24 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
20714 conjugacy classes of subgroups; only the 2178 normal subgroups are shown.
This group has 20714 conjugacy classes of subgroups — too many to draw. Only its 2178 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 — 5 generators on 22 points:
(1,2);(3,4)(5,6)(7,8)(9,10)(11,12,13,14);(4,6)(15,16)(17,20)(18,19)(21,22);(8,10)(16,19)(20,22);(15,17,18,21)(16,20,19,22)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#517592, C2×#763336, C2×#609884, … |
| maximal normal subgroup of | C2×#517592, C2×#763336, C2×#609884, … |
| maximal quotient of | D3×#13851, C2×#154123, C2×#154320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421214);