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×#8170 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4,6,5)(16,20)(17,18)(21,22);(7,8,9,11,10,12,13,14)(15,16,18,22,19,21,17,20);(3,5,6,4)(16,21)(20,22)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72211620, C22×#8170, C2×#182258, … |
| maximal normal subgroup of | #72211620, C22×#8170, C2×#182258, … |
| maximal quotient of | C22×#8170, C2×#128555, C2×#96686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89339);