Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,11,16,12,8,10,15)(7,17,14,13,20,9,19,18);(1,3,4,2)(5,7)(6,13)(8,18)(9,15)(10,14)(11,19)(12,20)(16,17);(5,8)(6,12)(7,18)(9,19)(10,16)(11,15)(13,20)(14,17);(5,9,11,18,12,17,10,13)(6,14,16,20,8,19,15,7)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,e^b=eh,g^b=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23049, #6382005, #6382007, … |
| maximal normal subgroup of | C2×#23049, #6382005, #6382007, … |
| maximal quotient of | #229187, #229193, #229197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16037);