Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#10663 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #399 |
| prime radical | C24 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,8,10,7,5,6,9)(11,12,16,24,15,19,14,18)(13,20,21,17,23,26,22,25);(3,5,7,4)(6,9,8,10)(11,13,14,21,15,23,16,22)(12,17,18,26,19,25,24,20);(3,4,7,5)(6,10,8,9)(11,14,15,16)(12,18,19,24)(13,21,23,22)(17,20,25,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151024);