Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 2113929216 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C212 |
| direct product | C23×#95 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C26 | C22 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(3,4);(5,6);(1,2)(3,4)(5,6);(12,14)(15,16,18,17);(7,8,9,10)(11,12,13,14)(15,17,18,16)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26974);