Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,17,21)(13,18,15,19)(14,23,26,22)(16,25,24,20);(3,5)(4,7)(6,10)(8,9)(11,13,17,15)(12,18,21,19)(14,20,26,25)(16,22,24,23);(11,14,24,19,17,26,16,18)(12,15,25,22,21,13,20,23);(5,10)(7,9)(11,15,17,13)(12,19,21,18)(14,25,26,20)(16,23,24,22);(4,8)(7,9)(13,15)(14,26)(18,19)(22,23)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484846);