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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31);(9,11,13,16)(10,14,12,15)(19,26)(21,29)(23,25)(24,28);(1,2)(3,7)(4,8)(5,6)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,14,16,15)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(1,4,5,3)(2,8,6,7)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,d^2=fg,e^2=g,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×#44691, #7668883, #7669111, … |
| maximal normal subgroup of | C2×#44691, #7668883, #7669111, … |
| maximal quotient of | #535483, #535768, #536151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37679);