Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#576 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,10,14)(7,16,12,19)(8,17,18,11)(9,15,20,13);(5,7)(6,8)(9,17)(10,12)(11,20)(13,19)(14,18)(15,16);(2,4)(5,8,10,18)(6,11,14,17)(7,13,12,15)(9,16,20,19);(5,8,10,18)(6,12,14,7)(9,19,20,16)(11,15,17,13)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24459, #6437902, #6437903, … |
| maximal normal subgroup of | C2×#24459, #6437902, #6437903, … |
| maximal quotient of | #268628, #268639, #268643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17447);