Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#10931 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8,9,10)(15,16)(17,19)(18,21)(20,22);(11,12,13,14)(16,19)(17,22)(20,21);(3,4,5,6)(8,10)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#364376, C2×#375848, C2×#354257, … |
| maximal normal subgroup of | C2×#364376, C2×#375848, C2×#354257, … |
| maximal quotient of | C2×#147466, C2×#147446, #72319995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261281);