Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C43 |
| direct product | C4×#1297 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1544 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 1544 conjugacy classes of subgroups — too many to draw. Only its 306 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4);(5,6,8,11,9,12,7,10)(13,14)(15,19)(16,18)(17,20);(6,10,12,11)(13,15)(14,17,18,19)(16,20)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12608);