Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26,30,31)(27,32,34,29)(28,36,37,33)(35,38,40,39);(1,2,4,5)(3,6,8,7)(10,14)(15,17)(16,21)(22,23)(26,31)(27,34)(28,37)(38,39);(2,5)(6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,37)(26,33,31,36)(27,35,34,40)(29,38,32,39);(1,3,4,8)(2,6,5,7)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,32)(26,27,31,34)(28,38,37,39)(33,35,36,40)Polycyclic presentation:
2^8|a^2=fg,b^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52794, #8092543, #8092651, … |
| maximal normal subgroup of | C2×#52794, #8092543, #8092651, … |
| maximal quotient of | #582338, #582509, #582555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45782);