Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C8 |
| direct product | C22×#1715 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C22×#493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3130 conjugacy classes of subgroups; only the 818 normal subgroups are shown.
This group has 3130 conjugacy classes of subgroups — too many to draw. Only its 818 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 — 5 generators on 20 points:
(1,2);(3,4);(6,9)(7,10)(11,12)(14,17)(15,19)(18,20);(5,6,10,12,8,11,7,9);(6,11)(9,12)(13,14,15,17,16,18,19,20)104 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#19745, #72214581, #72219583, … |
| maximal normal subgroup of | C22×#19745, #72214581, #72219583, … |
| maximal quotient of | C2×D3×#1715 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264280);