Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 131072 |
| 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 | |
655 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)(11,12)(13,21)(14,15)(16,19)(17,22)(18,24)(20,23)(25,26)(27,32)(28,29)(30,38)(31,34)(33,39)(35,36)(37,40);(2,3)(4,8)(10,15)(12,19)(17,21)(20,24)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,40,38,36);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,24)(22,23)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,38)(39,40);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,29,31,27)(26,32,34,28)(30,37,39,35)(33,40,38,36);(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,20,19,24)(25,30,31,39)(26,33,34,38)(27,35,29,37)(28,36,32,40)Polycyclic presentation:
2^8|a^2=h,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 | #7223740, #7223744, #7223770, … |
| maximal normal subgroup of | #7223740, #7223744, #7223770, … |
| maximal quotient of | #582269, #582445, #582453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46135);