Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C4×#364 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C4⋊C8 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,5,6);(7,8,11,16,12,15,20,22,13,17,19,14,10,21,9,18);(7,9,10,19,13,20,12,11)(8,14,15,18,17,16,21,22)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×#1369, C2×C4×#1336, C2×C4×#2171, … |
| maximal normal subgroup of | C2×C4×#1369, C2×C4×#1336, C2×C4×#2171, … |
| maximal quotient of | C2×C8×#364 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400293);