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 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,14)(11,17)(16,21)(18,23)(25,26,30,33)(27,28,35,31)(29,32,37,39)(34,36,40,38);(1,3,5,8)(2,6,7,4)(10,14)(15,20)(16,21)(22,24)(29,37)(32,39)(34,40)(36,38);(2,7)(4,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(1,4,5,6)(2,3,7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,31)(26,27,33,35)(29,36,37,38)(32,34,39,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=fgh,d^2=g,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×#57109, #8099664, #8100305, … |
| maximal normal subgroup of | C2×#57109, #8099664, #8100305, … |
| maximal quotient of | #586229, #586414, #586479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50097);