Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C43 · Schur multiplier (H2) = C28×C43 |
| direct product | C2×C42×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C42 | C22 |
| commutator | C2 | C22×C43 |
| Frattini, cosocle | C24 | C25 |
| socle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
12090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6);(7,8,9,10);(11,12)(13,14)(15,16,17,18);(12,14)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C4×#2433, C2×C4×#2428, C2×C22⋊C4×#127, … |
| maximal normal subgroup of | C2×C4×#2433, C2×C4×#2428, C2×C22⋊C4×#127, … |
| maximal quotient of | #72309033, C2×C22⋊C4×#95, C22×C42×C22⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420518);