Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 4718592 |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C22×C8×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C22 |
| commutator | C2 | C24×C8 |
| Frattini, cosocle | C2×C4 | C25 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,7,8,9,10,11,12);(13,14,15,16)(17,20,19,18);(13,17,15,19)(14,18,16,20)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53173);