Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×#7078 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| prime radical | C25 | D8 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
744 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 744 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,9,10,7)(6,8)(12,16)(13,19)(14,21)(17,26)(18,25)(20,23)(22,24);(4,7)(8,9)(11,12,15,18)(13,20,22,26)(14,23,21,17)(16,24,25,19)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29733);