Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C22×C4×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C4 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C24 | C22×D4 |
| prime radical | C23×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
12770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(9,10)(11,13)(12,14)(15,16);(10,13)(11,16)(14,15);(10,14)(13,15)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6432606, C22×#61451, C2×C4×#3003, … |
| maximal normal subgroup of | C2×#6432606, C22×#61451, C2×C4×#3003, … |
| maximal quotient of | C22×C4×#442, C22×#21503, C2×C4×#3003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481334);