Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13001 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1648 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)(7,8,10,14,11,15,20,21,12,16,9,17,19,22,18,13);(8,13,16,21)(9,18)(10,20)(11,19)(14,15,17,22);(4,6)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#495527, C2×#495949, C2×#495725, … |
| maximal normal subgroup of | C2×#495527, C2×#495949, C2×#495725, … |
| maximal quotient of | D3×#13001, #72225030, #72231439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401662);