Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(11,17)(15,20)(18,23)(22,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,40)(37,39);(2,5)(6,7)(10,14)(12,19)(15,20)(18,23)(25,27,30,28)(26,31,33,35)(29,34,37,36)(32,38,39,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,30,27)(26,31,33,35)(29,36,37,34)(32,38,39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,29,30,37)(26,32,33,39)(27,34,28,36)(31,38,35,40);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,26)(27,35)(28,31)(29,32)(30,33)(34,40)(36,38)(37,39)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | #98128, #179289, #179290, … |
| maximal normal subgroup of | #98128, #179289, #179290, … |
| maximal quotient of | #606827, #606834, #606835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46356);