Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2282 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);(3,4,7,8)(5,9,6,10)(11,12)(13,16)(14,18)(15,17);(4,8)(9,10)(11,13,15,14)(12,16,17,18)(19,20,22,25)(21,23,26,24);(3,5,7,6)(4,9,8,10)(11,14,15,13)(12,16,17,18)(20,23)(21,26)(24,25);(3,6,7,5)(4,9,8,10)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355358);