Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C4×#1789 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(5,6)(7,8)(9,10,12,15,13,11,16,14);(9,11)(10,13)(12,14)(15,16);(6,8)152 conjugacy classes — view the full character table.
| maximal subgroup of | #72239472, C4×#32888, AGL(1,𝔽5)×#1789 |
| maximal normal subgroup of | #72239472, C4×#32888 |
| maximal quotient of | C4×#8440, AGL(1,𝔽5)×#1789 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262010);