Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13408 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1324 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)(5,6)(8,9)(11,12,17,23)(13,18,22,16)(14,19,15,24)(20,26,21,25);(7,8)(9,10)(11,13)(12,18)(14,20)(15,21)(16,23)(17,22)(19,26)(24,25);(4,6)(7,9,10,8)(12,19)(13,21)(14,25)(15,26)(16,18)(20,22)(23,24)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406256);