Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 | #3522 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
903 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)(6,8)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,15)(13,22)(14,20)(17,18)(19,21)(23,24);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)Polycyclic presentation:
2^8|a^2=h,c^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #221353, #221363, #221373, … |
| maximal normal subgroup of | #221353, #221363, #221373, … |
| maximal quotient of | #6379569, #6379570, #6380397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54487);