Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#7890 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2442 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)(5,6)(7,8,10,13,11,14,9,12)(15,16,18,21,20,22,19,17);(4,6)(8,12,14,13)(15,17,19,22,20,21,18,16);(7,9,11,10)(8,12,14,13)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88154);