Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,7)(9,10,13,19)(11,20,22,15)(12,17,14,18)(16,23,24,21);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,23)(13,22)(14,21)(16,17)(18,24)(19,20);(9,11)(10,16)(12,21)(13,22)(14,23)(15,17)(18,20)(19,24);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,17,19,18)(11,21,22,23)(15,24,20,16);(9,12,13,14)(10,18,19,17)(11,21,22,23)(15,16,20,24)Polycyclic presentation:
2^8|a^2=h,b^2=fh,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,e^b=eh,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #250082, #250114, #250146, … |
| maximal normal subgroup of | #250082, #250114, #250146, … |
| maximal quotient of | #6379403, #6379471, #6380340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54336);