Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12929 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1278 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)(7,8,10,15,12,16,11,21)(9,17,19,22,20,13,18,14);(8,13,16,17)(9,18)(10,11)(14,15,22,21)(19,20);(4,6)92 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#495476, C2×#495949, C2×#495725, … |
| maximal normal subgroup of | C2×#495476, C2×#495949, C2×#495725, … |
| maximal quotient of | D3×#12929, #72224980, #72231314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401460);