Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,14,17)(11,15,18,23)(12,13,16,22)(19,24,21,20)(25,26,31,33)(27,34,28,29)(30,38,39,32)(35,37,36,40);(11,18)(12,16)(13,22)(15,23)(25,27,31,28)(26,29,33,34)(30,35,39,36)(32,37,38,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,20)(14,18)(16,19)(17,23)(22,24)(25,28,31,27)(26,29,33,34)(30,36,39,35)(32,37,38,40);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,13,17,22)(11,19,18,21)(15,24,23,20)(25,29)(26,27)(28,33)(30,37)(31,34)(32,35)(36,38)(39,40);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,20)(12,17)(14,22)(15,19)(18,24)(21,23)(25,30)(26,32)(27,35)(28,36)(29,37)(31,39)(33,38)(34,40)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,d^2=gh,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50997, #7706655, #7706726, … |
| maximal normal subgroup of | C2×#50997, #7706655, #7706726, … |
| maximal quotient of | #560840, #561129, #561960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43985);