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×#62313 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,16)(14,22)(15,19)(17,21)(18,25)(20,23)(24,26);(3,5)(4,7)(6,10)(8,9)(11,13,16,12)(14,20,24,18)(15,19,17,21)(22,25,26,23);(11,14,17,25,16,24,15,23)(12,18,19,22,13,20,21,26);(5,10)(7,9)(11,15,16,17)(12,19,13,21)(14,23,24,25)(18,22,20,26);(4,8)(7,9)(12,13)(14,24)(19,21)(23,25)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484805);