Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C8 |
| direct product | C4×#1712 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#127 |
| prime radical | C2×#1712 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1268 conjugacy classes of subgroups; only the 334 normal subgroups are shown.
This group has 1268 conjugacy classes of subgroups — too many to draw. Only its 334 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(6,9)(7,10)(11,12)(14,17)(15,18)(19,20);(5,6,10,12,8,11,7,9);(13,14,18,20,16,19,15,17)152 conjugacy classes — view the full character table.
| maximal subgroup of | #72216165, #72216170, #72216171, … |
| maximal normal subgroup of | #72216165, #72216170, #72216171, … |
| maximal quotient of | #72216165, #72216249, AGL(1,𝔽5)×#1712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120083);