Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C213 |
| direct product | C22×#1779 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C25 | C22×C4 |
| prime radical | C27 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
30426 conjugacy classes of subgroups; only the 3058 normal subgroups are shown.
This group has 30426 conjugacy classes of subgroups — too many to draw. Only its 3058 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 — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,8)(9,10)(11,12)(13,14,15,17,16,18,19,20);(6,8);(10,12)224 conjugacy classes — view the full character table.
| maximal subgroup of | #72268039, #72268040, #72268048, … |
| maximal normal subgroup of | #72268039, #72268040, #72268048, … |
| maximal quotient of | C2×#359422, C2×#184843, C2×#493333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277875);