Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
| direct product | C2×#10737 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#95 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(7,8)(9,12,15,19)(10,17)(11,13)(14,18)(16,22,21,20);(7,9,10,16,11,15,18,21)(8,12,14,20,13,19,17,22);(4,6)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#364416, C2×#374114, C2×#394040, … |
| maximal normal subgroup of | C2×#364416, C2×#374114, C2×#394040, … |
| maximal quotient of | D3×#10737, #72312171, C2×#123516 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260607);