Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C215 |
| direct product | C22×C4×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| socle | C25 | C24 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
20258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(10,12)(14,16);(13,14,15,16);(9,10)(11,12)224 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#21452, C22×#34290, C22×#36096, … |
| maximal normal subgroup of | C22×#21452, C22×#34290, C22×#36096, … |
| maximal quotient of | C22×#21503, C22×#21546, #72261526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532444);