Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(11,15)(14,16)(17,18,22,23)(19,21,25,31)(20,27,30,24)(26,32,28,29);(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,14,15,16)(18,23)(21,31)(24,27)(29,32);(2,3)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19,22,25)(18,21,23,31)(20,28,30,26)(24,32,27,29);(1,3,5,2)(4,7,8,6)(9,12,13,10)(11,16,15,14)(17,20,22,30)(18,24,23,27)(19,26,25,28)(21,29,31,32);(1,4)(2,6)(3,7)(5,8)(17,21)(18,25)(19,23)(20,29)(22,31)(24,28)(26,27)(30,32)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#41494, #7639714, #7639806, … |
| maximal normal subgroup of | C2×#41494, #7639714, #7639806, … |
| maximal quotient of | #544647, #545176, #545611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34482);