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 | |
663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,14)(13,20)(16,21)(17,23)(18,19)(22,24)(27,31)(28,35)(29,37)(32,39);(2,3)(4,8)(10,12)(15,19)(16,20)(22,23)(25,26,30,33)(27,28,31,35)(29,32,37,39)(34,36,38,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27)(26,28)(29,34)(30,31)(32,36)(33,35)(37,38)(39,40);(1,4,5,8)(2,6,3,7)(9,12,14,10)(11,15,18,19)(13,20,21,16)(17,22,24,23)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(9,13,14,21)(10,16,12,20)(11,17,18,24)(15,22,19,23)(25,29,30,37)(26,32,33,39)(27,34,31,38)(28,36,35,40)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=h,d^2=f,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 | #7232346, #7232347, #7232350, … |
| maximal normal subgroup of | #7232346, #7232347, #7232350, … |
| maximal quotient of | #582228, #582439, #582616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46131);