Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#7866 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2054 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)(13,19)(14,22)(15,18)(16,21)(17,23)(20,25)(24,26);(11,13)(12,16,18,24)(14,21)(15,20)(17,19,22,25)(23,26);(13,20)(16,24)(19,25)(21,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88091);