Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
| direct product | C2×#34312 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C22×#548 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11246 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);(3,4,6,5)(8,10)(11,12,17,15)(13,20,14,18)(16,23,24,19)(21,25,22,26);(11,13,17,14)(12,18,15,20)(16,21,24,22)(19,25,23,26);(7,8,9,10)(11,14,17,13)(12,18,15,20)(16,22,24,21)(19,25,23,26);(11,15,17,12)(13,18,14,20)(16,23,24,19)(21,25,22,26);(3,5,6,4)(11,16,17,24)(12,19,15,23)(13,21,14,22)(18,25,20,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532808);