Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = C2×C8×#120 (order 512) · Inn(G) = C22⋊C4 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C16 |
| Frattini, cosocle | C4×C16 | C22 |
| socle, prime radical | C22 | C22⋊C16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,12,18,14,19,25,33,15,16,23,28,27,29,36,38,13,17,24,30,26,31,37,39,11,20,21,32,22,34,35,40);(1,2,4,6)(3,8,7,5)(9,11,13,15)(10,16,17,20)(12,23,24,21)(14,22,26,27)(18,32,30,28)(19,29,31,34)(25,36,37,35)(33,40,39,38)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=eh,d^2=f,f^2=g,g^2=h,b^a=bc,c^a=ce,c^b=ceh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7393, #150080, #150081, … |
| maximal normal subgroup of | C2×#7393, #150080, #150081, … |
| maximal quotient of | #93927, #93967, #94039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 381);