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×#62343 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3548 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,17,21)(13,23,24,19)(14,16,22,18)(15,20,26,25);(3,5)(4,7)(6,10)(8,9)(11,13)(12,19)(14,25)(15,18)(16,26)(17,24)(20,22)(21,23);(11,14,18,12,17,22,16,21)(13,20,15,23,24,25,26,19);(5,10)(7,9)(11,15)(12,20)(13,16)(14,19)(17,26)(18,24)(21,25)(22,23);(4,8)(7,9)(11,16,17,18)(12,14,21,22)(13,15,24,26)(19,20,23,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484832);