Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | SD16×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C24×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | D4×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2,3,4,5,6,7,8);(2,4)(3,7)(6,8);(9,10)(11,15)(12,13)(14,16);(10,13)(14,15);(9,10)(11,14)(12,13)(15,16);(9,11)(10,14)(12,16)(13,15)119 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10490210);