Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19078 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3080 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);(4,7)(5,9)(8,10)(11,12,17,21)(13,22,25,26)(14,24,18,23)(15,20,16,19);(5,9)(7,10)(11,13)(12,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,22);(3,4)(5,10)(6,8)(7,9)(11,14,17,18)(12,13,21,25)(15,26,16,22)(19,24,20,23);(11,15,17,16)(12,19,21,20)(13,23,25,24)(14,22,18,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258686);