Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26405 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2125 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);(3,4,7,9)(5,8,6,10)(11,12,17,21)(13,23,19,25)(14,24,15,18)(16,22,26,20);(3,5,7,6)(4,8,9,10)(11,13)(12,15)(14,21)(16,18)(17,19)(20,25)(22,23)(24,26);(3,6,7,5)(4,8,9,10)(11,14,17,15)(12,18,21,24)(13,22,19,20)(16,25,26,23);(11,15,17,14)(12,19,21,13)(16,25,26,23)(18,20,24,22)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346446);