Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
| direct product | D8×#97 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,9)(8,11)(10,12);(6,9)(7,12)(10,11);(13,14,15,18,16,17,19,20);(1,2,3,4)(14,17)(18,20)140 conjugacy classes — view the full character table.
| maximal subgroup of | D8×#442, #72279416, #97×AGL(1,ℤ/8), … |
| maximal normal subgroup of | D8×#442, #72279416, #97×AGL(1,ℤ/8), … |
| maximal quotient of | D8×#347 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91187);