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) = C25×C4 |
| direct product | M16×#102 |
| 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 | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,7,10,8,9,11,12);(6,9)(10,12);(1,2,3,4)(13,14)(15,17)(16,19)(18,20);(14,17)(15,20)(18,19)140 conjugacy classes — view the full character table.
| maximal subgroup of | M16×#442, #102×AGL(1,ℤ/8), #72279582 |
| maximal normal subgroup of | M16×#442, #102×AGL(1,ℤ/8), #72279582 |
| maximal quotient of | #72310117, M16×#360, M16×#361 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90372);