Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 65536 |
| 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 | |
679 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(28,36)(31,38)(34,40)(37,39);(2,4)(3,7)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3,5,7)(2,6,4,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,21)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,4,5,2)(3,8,7,6)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=f,d^2=fg,e^2=g,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 | #132964, #487121, #487249, … |
| maximal normal subgroup of | #132964, #487121, #487249, … |
| maximal quotient of | #582250, #582471, #582658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45896);