Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C4×#441 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle, prime radical | C24 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,5,6);(7,8,10,9)(11,12)(13,19)(14,22)(15,17)(16,20)(18,24)(21,26)(23,25);(11,13,15,21)(12,16,17,25)(14,23,24,20)(18,26,22,19);(7,9,10,8)(12,17)(16,25)(18,22)(19,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277253);