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×#62281 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3536 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6690 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)(11,12,16,19)(13,21,23,18)(14,15,20,17)(22,26,25,24);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,25)(15,26)(16,23)(17,24)(19,21)(20,22);(11,14,17,12,16,20,15,19)(13,22,24,21,23,25,26,18);(5,10)(7,9)(13,23)(18,21)(22,25)(24,26);(4,8)(7,9)(11,15,16,17)(12,14,19,20)(13,24,23,26)(18,22,21,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484776);