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) = C24×C4 |
| direct product | M16×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,6,4,5,7,8);(2,5)(6,8);(9,10,12,15,13,11,16,14);(9,11)(10,13)(12,14)(15,16)140 conjugacy classes — view the full character table.
| maximal subgroup of | #1042, M16×#857, #72226235, … |
| maximal normal subgroup of | #1042, #72226235, #72226238, … |
| maximal quotient of | #72239508 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90350);