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 | |
714 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26,29,30)(27,31,33,37)(28,32,35,38)(34,39,36,40);(1,3,6,5)(2,7,4,8)(10,14)(15,21)(16,19)(22,23)(26,30)(27,33)(28,35)(39,40);(1,4,6,2)(3,7,5,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,27)(26,31)(28,36)(29,33)(30,37)(32,40)(34,35)(38,39);(1,5,6,3)(2,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,28)(26,32)(27,34)(29,35)(30,38)(31,39)(33,36)(37,40);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fh,d^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×#56720, #8099728, #8100298, … |
| maximal normal subgroup of | C2×#56720, #8099728, #8100298, … |
| maximal quotient of | #585895, #585903, #586540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49708);