Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,15)(11,16,17,12)(13,21,22,18)(19,20,24,23)(25,26)(27,32)(28,29)(30,38)(31,34)(33,39)(35,36)(37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(12,16)(18,21)(20,23)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,40,38,36);(2,5)(6,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,21)(22,24)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,29,31,27)(26,32,34,28)(30,37,39,35)(33,40,38,36);(1,3)(2,6)(4,7)(5,8)(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=fg,e^2=f,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 | C2×#52521, #8092391, #8092468, … |
| maximal normal subgroup of | C2×#52521, #8092391, #8092468, … |
| maximal quotient of | #582323, #582523, #582556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45509);