Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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:
(1,2,6,4)(3,7,5,8)(11,15)(14,16)(17,18,22,25)(19,21,24,30)(20,23,29,32)(26,28,27,31);(2,4)(3,5)(9,10,13,12)(11,14,15,16)(19,24)(21,30)(26,27)(28,31);(1,3,6,5)(2,7,4,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,24)(18,21,25,30)(20,27,29,26)(23,31,32,28);(1,4,6,2)(3,7,5,8)(17,20)(18,23)(19,26)(21,28)(22,29)(24,27)(25,32)(30,31);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,25)(20,28)(22,30)(23,27)(26,32)(29,31)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47551, #7700866, #7700916, … |
| maximal normal subgroup of | C2×#47551, #7700866, #7700916, … |
| maximal quotient of | #577016, #577019, #577152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40539);