Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#11452 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| socle | C23 | C2×#99 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,17)(14,21,22,16)(15,18)(19,25,26,23)(20,24);(12,16,18,21)(14,22)(17,23,24,25)(19,26);(11,13,15,20)(12,17,18,24)(14,19,22,26)(16,23,21,25)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387232);