Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| 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 | #3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(2,5)(3,8)(6,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23);(10,16)(14,17)(15,24)(22,23);(2,6)(5,7)Polycyclic presentation:
2^8|b^2=f,c^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eh,g^a=gh,c^b=cf,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #112149, #112166, #112181, … |
| maximal normal subgroup of | #112149, #112166, #112181, … |
| maximal quotient of | #6374388, #6374450, #6374983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53848);