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 | #2535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
837 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(19,23)(21,24);(2,6)(3,8)(10,12)(11,15)(17,18,20,22)(19,21,23,24);(9,11,13,15)(10,14,12,16)(17,19)(18,21)(20,23)(22,24);(1,3,5,8)(2,7,6,4)(9,11)(10,14)(12,16)(13,15);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,16,15,14)Polycyclic presentation:
2^8|b^2=g,c^2=h,d^2=fh,e^2=f,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×#44791, #7669172, #7669234, … |
| maximal normal subgroup of | C2×#44791, #7669172, #7669234, … |
| maximal quotient of | #535367, #535687, #535996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37779);