Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62094 |
| 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 | |
6666 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)(11,12,17,22)(13,18,14,24)(15,21,16,20)(19,23,26,25);(3,4)(5,8)(6,7)(9,10)(11,13,17,14)(12,18,22,24)(15,25,16,23)(19,20,26,21);(11,14,17,13)(12,19,22,26)(15,25,16,23)(18,20,24,21);(4,7)(8,10)(11,15,17,16)(12,20,22,21)(13,23,14,25)(18,19,24,26);(3,5,6,9)(4,8,7,10)(11,15,17,16)(12,21,22,20)(13,23,14,25)(18,26,24,19)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486086);