Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C211×C4 |
| direct product | C22×C4⋊#95 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C26 | C23 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
15026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 20 points:
(3,4);(1,2)(3,4);(6,8)(10,11)(17,18,20,19);(9,10,12,11)(13,14,15,16)(17,19,20,18);(5,6,7,8)(9,11,12,10)224 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#14806, C22×#13899, C22×#14698, … |
| maximal normal subgroup of | C22×#14806, C22×#13899, C22×#14698, … |
| maximal quotient of | C23×C4⋊#95, C2×#153831, C22×#7557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420533);