Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C8 |
| direct product | C2×#8148 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1546 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);(3,4)(5,6)(7,8,10,13,11,9,12,14)(15,16,19,17,18,21,20,22);(4,6)(15,16,20,22,18,21,19,17);(7,9,10,14,11,8,12,13)(15,17,19,21,18,22,20,16)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89086);