Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C4×#2358 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6614 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);(6,8)(9,10)(11,12)(13,14,16,15);(5,6)(7,8)(13,15,16,14);(10,12);(6,8)200 conjugacy classes — view the full character table.
| maximal subgroup of | #72278128, C42×D42, C4×#58990, … |
| maximal normal subgroup of | #72278128, C42×D42, C4×#58990, … |
| maximal quotient of | #72278186, #72299186, #72299697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420716);