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×#34266 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11218 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);(4,6)(7,8,10,9)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24);(7,8,10,9)(12,14)(17,19)(18,22)(24,25);(3,4)(5,6)(7,9,10,8)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25);(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24);(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532747);