Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7457 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×#7457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1368 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | #346 | C23 |
| prime radical | C23 | #365 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(4,9,10,19,12,21,24,22)(5,13,14,27)(6,16)(7,18)(11,25,23,20,26,33,32,34)(15,30,29,28)(17,31);(3,4,5,9,6,11,13,20,7,10,15,22,16,23,28,25,8,12,14,21,17,26,27,34,18,24,29,19,31,32,30,33)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216304, C2×AGL(1,ℤ/32), #72216321, … |
| maximal normal subgroup of | #72216304, C2×AGL(1,ℤ/32), #72216321, … |
| maximal quotient of | D3×#7457, C4×#7457, C22×#7457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58219);