Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7515 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#7518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1402 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | #492 | C23 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ #492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 50 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);(3,4,6,11,8,13,20,29)(5,14,15,27,17,24,31,34)(7,10,18,26,21,12,19,28)(9,22,23,32,25,30,33,16);(3,5,8,17)(4,9,13,25)(6,15,20,31)(7,16,21,32)(10,14,12,24)(11,27,29,34)(18,22,19,30)(23,26,33,28)44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#153640, C2×#153628, C2×#153642, … |
| maximal normal subgroup of | C2×#153640, C2×#153628, C2×#153642, … |
| maximal quotient of | #72283057 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60636);