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 | |
711 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)(3,8,4,6)(9,10)(11,17)(12,19)(13,21)(14,15)(16,22)(18,20)(23,24)(25,26)(27,32)(28,34)(29,31)(30,33)(35,39)(36,40)(37,38);(1,3)(2,6)(4,5)(7,8)(10,15)(12,17)(13,22)(18,24)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(3,4)(6,8)(9,11,14,19)(10,12,15,17)(13,18,22,24)(16,20,21,23)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,4,5,3)(2,6,7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,2)(3,8)(4,6)(5,7)(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=g,d^2=fg,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×#56247, #8099917, #8100299, … |
| maximal normal subgroup of | C2×#56247, #8099917, #8100299, … |
| maximal quotient of | #586205, #586673, #586795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49235);