Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C4×#1867 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2270 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);(5,6,8,7)(13,16);(5,7,8,6)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,15)(14,16)128 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#1867, #72311708, C4×#24404, … |
| maximal normal subgroup of | D4×#1867, #72311708, C4×#24404, … |
| maximal quotient of | #72311919, #72312075, #72312494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261652);