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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(26,30)(27,33)(28,36)(38,40);(2,3)(6,7)(10,13)(11,17)(12,20)(23,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(9,11,14,17)(10,15,13,19)(12,21,20,18)(16,24,22,23)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,3,5,2)(4,7,8,6)(9,12,14,20)(10,16,13,22)(11,18,17,21)(15,23,19,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,19,17,15)(12,22,20,16)(18,24,21,23)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=h,d^2=fg,e^2=fg,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×#57066, #8099569, #8100308, … |
| maximal normal subgroup of | C2×#57066, #8099569, #8100308, … |
| maximal quotient of | #586664, #586711, #587327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50054);