Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C210×C4 |
| direct product | C4×D4×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
9210 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);(5,6)(7,8);(6,8);(9,10)(11,12)(13,14,15,16);(10,12)200 conjugacy classes — view the full character table.
| maximal subgroup of | D42×C22⋊C4, D4×#2433, C4×D4×#120, … |
| maximal normal subgroup of | D42×C22⋊C4, D4×#2433, C4×D4×#120, … |
| maximal quotient of | #72224693, #72265213, #72265244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420696);