Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24765 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,5)(11,12,16,21,17,22,15,20)(13,18,23,19,14,24,25,26);(3,4,6,5)(7,8)(9,10)(11,13)(12,18)(14,17)(15,25)(16,23)(19,21)(20,26)(22,24);(8,10)(11,14,17,13)(12,19,22,26)(15,25,16,23)(18,20,24,21);(3,5,6,4)(11,15,17,16)(12,20,22,21)(13,23,14,25)(18,19,24,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350173);