Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#23020 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4)(5,6)(8,12,14,18)(13,15,20,21);(7,8,10,18,11,14,19,12)(9,15,17,13,16,21,22,20);(4,6)(9,16)(13,20)(15,21)(17,22);(7,9,10,17,11,16,19,22)(8,13,12,15,14,20,18,21)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289105);