Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(6,8)(11,15)(14,16);(1,2,3,4)(9,10)(11,16)(12,13)(14,15);(5,6)(7,8)(10,13)(14,16);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|b^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #132517, #132569, #487299, … |
| maximal normal subgroup of | #132517, #132569, #487299, … |
| maximal quotient of | #535285, #535921, #536489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37858);