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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,24)(20,23)(25,26,30,33)(27,31,28,35)(29,37,38,32)(34,40,36,39);(1,2)(3,5)(4,6)(7,8)(10,15)(12,19)(17,21)(20,24)(25,27,30,28)(26,31,33,35)(29,34,38,36)(32,39,37,40);(1,2,4,6)(3,5,7,8)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,24)(22,23)(25,26)(27,31)(28,35)(29,32)(30,33)(34,39)(36,40)(37,38);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,28,30,27)(26,31,33,35)(29,36,38,34)(32,39,37,40);(1,3,4,7)(2,5,6,8)(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,20,19,24)(25,29,30,38)(26,32,33,37)(27,34,28,36)(31,39,35,40)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,d^2=g,e^2=fgh,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 | #7228604, #7228608, #7228612, … |
| maximal normal subgroup of | #7228604, #7228608, #7228612, … |
| maximal quotient of | #582269, #582455, #582731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46119);