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×#62438 |
| 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 | |
7018 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)(13,22)(14,25)(15,20)(16,21)(17,23)(18,19)(24,26);(3,4)(5,8)(6,7)(9,10)(11,13)(12,18)(14,23)(15,25)(16,19)(17,24)(20,26)(21,22);(11,14,17,18,16,26,15,22)(12,19,20,25,21,13,23,24);(4,7)(8,10)(11,15,16,17)(12,20,21,23)(13,24,19,25)(14,22,26,18);(3,5,6,9)(4,8,7,10)(12,21)(13,19)(20,23)(24,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486400);