Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | D4×#496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,3,4);(2,4);(6,8)(10,13)(11,15)(14,16);(9,10,11,13,12,14,15,16);(5,6,7,8)(10,14)(13,16)80 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#3264, D4×#3399, D4×#3180 |
| maximal normal subgroup of | D4×#3264, D4×#3399, D4×#3180 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490377);