Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×#1422 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C16 | D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×C4⋊C16 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4);(5,6,9,15,10,16,24,31,11,14,25,29,21,30,34,36,12,17,26,32,19,13,27,28,8,22,23,35,7,18,20,33);(5,7,8,19,12,21,11,10)(6,13,14,18,17,16,22,30)(9,20,23,27,26,34,25,24)(15,28,29,33,32,31,35,36)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59173);