Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C42 · Schur multiplier (H2) = C214×C4 |
| direct product | C22×C42×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C42 | C22 |
| commutator | C2 | C24×C42 |
| Frattini, cosocle | C23 | C26 |
| socle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
23178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2);(3,4);(5,6,7,8);(9,10,11,12);(13,14,15,16);(14,16)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C42×#127, C22×#8170, C2×C4×#1955, … |
| maximal normal subgroup of | C2×C42×#127, C22×#8170, C2×C4×#1955, … |
| maximal quotient of | C2×C42×#127, C22×C4×#442, C4×#60182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532399);