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×#62466 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6980 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);(4,7)(5,9)(11,12,16,20)(13,22,23,18)(14,24,25,26)(15,21,17,19);(3,4)(5,8)(6,7)(9,10)(11,13,16,23)(12,18,20,22)(14,21,25,19)(15,26,17,24);(11,14,17,18,16,25,15,22)(12,13,21,24,20,23,19,26);(4,7)(8,10)(13,23)(14,25)(18,22)(24,26);(3,5,6,9)(4,8,7,10)(11,15,16,17)(12,19,20,21)(13,24,23,26)(14,18,25,22)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486421);