Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×C42×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C22×D4 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4194 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);(12,18)(13,17)(14,16)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#2177, C2×C4×#1580, C2×C4×#1333, … |
| maximal normal subgroup of | C2×C4×#2177, C2×C4×#1580, C2×C4×#1333, … |
| maximal quotient of | C2×C4×#2177, C2×C4×C8×D8, C2×#493242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277247);