Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C26 |
| direct product | C2×#11613 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
994 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);(3,4,7,10,8,5,6,9)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(3,5)(4,8)(6,10)(7,9);(3,6,8,7)(4,9,5,10)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388284);