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 | |
751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,8,5,7)(10,13)(14,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31);(1,3,6,5)(2,7,4,8)(19,26)(21,29)(23,25)(24,28);(1,3,6,5)(2,8,4,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,4,6,2)(3,7,5,8)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(1,5,6,3)(2,7,4,8)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fgh,d^2=fg,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×#44561, #7668900, #7669096, … |
| maximal normal subgroup of | C2×#44561, #7668900, #7669096, … |
| maximal quotient of | #535569, #535734, #536259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37549);