Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#23172 |
| 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 | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,17,21,16,20,15,19)(13,22,23,26,24,18,14,25);(4,7)(5,9)(11,13,17,23,16,24,15,14)(12,18,21,25,20,22,19,26);(11,14,15,24,16,23,17,13)(12,18,21,25,20,22,19,26);(3,5,6,9)(4,8,7,10)(11,15,16,17)(12,19,20,21)(13,23,24,14)(18,25,22,26)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289909);