Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11242 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,6)(11,12,15,19,16,18,23,14)(13,20,21,17,22,25,26,24);(7,8)(9,10)(11,13,15,21,16,22,23,26)(12,17,14,20,18,24,19,25);(4,6)(8,10)(11,14,15,18,16,19,23,12)(13,20,21,24,22,25,26,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265256);