Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,14,15)(11,12,19,16)(13,21,22,17)(18,23,24,20)(25,26,31,33)(27,34,28,29)(30,38,39,32)(35,37,36,40);(1,2)(3,7)(4,5)(6,8)(10,15)(12,16)(13,22)(18,24)(25,27,31,28)(26,29,33,34)(30,35,39,36)(32,37,38,40);(2,5)(6,7)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,28,31,27)(26,29,33,34)(30,36,39,35)(32,37,38,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29)(26,27)(28,33)(30,37)(31,34)(32,35)(36,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,30)(26,32)(27,35)(28,36)(29,37)(31,39)(33,38)(34,40)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=f,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53885, #8103445, #8103476, … |
| maximal normal subgroup of | C2×#53885, #8103445, #8103476, … |
| maximal quotient of | #586402, #586690, #587070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46873);