Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C4×#3025 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #120 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4);(10,16)(17,20);(5,6)(7,10)(8,11)(9,12)(13,17)(14,16)(15,18)(19,20);(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20);(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,17,16,20)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279922);