Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 18874368 |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#2276 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,6,5)(7,8,10,9)(12,14)(16,17);(7,8,10,9)(11,12,15,14)(13,17,18,16);(3,5,6,4)(7,9,10,8)(11,13,15,18)(12,16,14,17);(11,14,15,12)(13,17,18,16)136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26986);