Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,7)(5,6)(9,10,14,15)(11,16,13,12)(18,21)(22,24);(1,3)(2,7)(4,6)(5,8)(10,15)(12,16);(3,4)(7,8)(9,11,14,13)(10,16,15,12)(17,18,20,21)(19,22,23,24);(1,4,6,3)(2,7,5,8)(9,12)(10,11)(13,15)(14,16);(1,5,6,2)(3,7,4,8)(9,13,14,11)(10,12,15,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=fh,c^2=fgh,d^2=h,e^2=fg,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 | C2×#44864, #7669285, #7669287, … |
| maximal normal subgroup of | C2×#44864, #7669285, #7669287, … |
| maximal quotient of | #535791, #535807, #536358, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37852);