Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62332 |
| 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 | |
6814 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)(12,18)(13,21)(15,17)(19,22)(23,24)(25,26);(3,5)(4,7)(6,10)(8,9)(11,12,16,18)(13,19,24,25)(14,17,20,15)(21,26,23,22);(11,13,15,23,16,24,17,21)(12,19,14,22,18,25,20,26);(5,10)(7,9)(11,14,16,20)(12,15,18,17)(13,22,24,26)(19,23,25,21);(4,8)(7,9)(11,15,16,17)(12,20,18,14)(13,23,24,21)(19,26,25,22)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484819);