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 | |
672 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)(9,10,14,15)(11,12,16,20)(13,21,22,17)(18,24,23,19)(25,26,30,33)(27,31,34,39)(28,29,32,38)(35,40,37,36);(1,3,5,2)(4,7,8,6)(10,15)(11,16)(13,22)(19,24)(27,34)(28,32)(29,38)(31,39);(1,4,5,8)(2,6,3,7)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,24)(22,23)(25,27)(26,31)(28,37)(29,36)(30,34)(32,35)(33,39)(38,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,21)(22,24)(25,28,30,32)(26,29,33,38)(27,35,34,37)(31,40,39,36);(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,19,20,24)(25,29)(26,32)(27,36)(28,33)(30,38)(31,35)(34,40)(37,39)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=g,d^2=fh,e^2=f,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×#50932, #7706701, #7706726, … |
| maximal normal subgroup of | C2×#50932, #7706701, #7706726, … |
| maximal quotient of | #560370, #560552, #561553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43920);