Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×C8×D16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D8 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C4×D8 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,5,6,7,8,9,10);(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)176 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#2177, C22×C8×D16, C2×C8×#509, … |
| maximal normal subgroup of | C8×#2177, C22×C8×D16, C2×C8×#509, … |
| maximal quotient of | C22×C8×D16, C2×C8×#357, C8×#2263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390688);