Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
831 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);(8,9)(11,12,13,15)(14,16,18,17);(3,4)(5,6)(7,8)(9,10)(12,15)(16,17);(4,6)(7,9,10,8)(12,16)(14,18)(15,17)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2036, #359140, #359142, … |
| maximal normal subgroup of | C4×#2036, #359140, #359142, … |
| maximal quotient of | C2×#10434, C2×#10436, C2×#10438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16466);