Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#1923 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C24 | C22×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,6)(8,9)(11,12,14,17,15,13,16,18);(7,8,10,9)(11,13,14,18,15,12,16,17);(4,6)(7,9,10,8)112 conjugacy classes — view the full character table.
| maximal subgroup of | #96681, #96683, #98828, … |
| maximal normal subgroup of | #96681, #96683, #98828, … |
| maximal quotient of | C2×#9541, C2×#9543, C2×#9542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14032);