Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7333 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×#7333 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #322 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | #346 | C23 |
| socle | C22 | #1313 |
| prime radical | C23 | #347 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 122 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);(3,4,5,10,6,12,16,26,7,11,17,28,19,23,33,34,8,13,15,27,18,30,32,24,20,29,14,25,31,9,21,22);(4,9,11,12,13,23,29,30)(5,14,15,17)(6,18)(10,24,25,34,27,26,28,22)(16,33,32,21)(19,31)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216250, C2×AGL(1,ℤ/32), #72216330, … |
| maximal normal subgroup of | #72216250, C2×AGL(1,ℤ/32), #72216330, … |
| maximal quotient of | #72217184, D3×#7333, C4×#7333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56699);