Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C27×C42 |
| direct product | C22⋊C4×#95 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C22 | C2×C43 |
| prime radical | C26 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
6938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(9,10,11,12)(18,20);(17,18)(19,20);(6,8)(13,14,16,15);(1,2,3,4)(5,6,7,8)(9,12,11,10)(13,15,16,14)200 conjugacy classes — view the full character table.
| maximal subgroup of | #72278186, #72278240, #72278310, … |
| maximal normal subgroup of | #72278186, #72278240, #72278310, … |
| maximal quotient of | #72299913, #72299915, C2×C22⋊C4×#95 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60919);