Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214×C4 |
| direct product | C23×#382 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C26 | C23 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
17122 conjugacy classes of subgroups; only the 10178 normal subgroups are shown.
This group has 17122 conjugacy classes of subgroups — too many to draw. Only its 10178 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);(8,9)(11,12,13,14)(17,21)(19,22);(7,8,10,9)(15,16,18,20)(17,19,21,22);(7,9,10,8)(15,17)(16,19)(18,21)(20,22)224 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#2328, C2×#153867, C2×#154074, … |
| maximal normal subgroup of | C23×#2328, C2×#153867, C2×#154074, … |
| maximal quotient of | C23×C4⋊#95, C22×#13881, C22×#14682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532423);