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×#26402 |
| 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)(11,12,16,20)(13,22,18,25)(14,23,24,17)(15,21,26,19);(3,5,7,6)(4,8,9,10)(11,13,16,18)(12,14,20,24)(15,23,26,17)(19,25,21,22);(3,6,7,5)(4,8,9,10)(11,14)(12,17)(13,21)(15,22)(16,24)(18,19)(20,23)(25,26);(11,14)(12,18)(13,20)(15,25)(16,24)(17,19)(21,23)(22,26)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346440);