Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#17518 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#3482 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,7,10,8,5,6,9)(11,12,17,14)(13,20,23,26)(15,19,16,24)(18,25,21,22);(3,5)(4,8)(6,10)(7,9)(11,13)(12,18)(14,21)(15,22)(16,25)(17,23)(19,26)(20,24);(3,6,8,7)(4,9,5,10)(11,14,17,12)(13,21,23,18)(15,19,16,24)(20,22,26,25);(3,6,8,7)(4,9,5,10)(11,15,17,16)(12,19,14,24)(13,22,23,25)(18,26,21,20)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250256);