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 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,17)(11,12,15,19)(13,16,21,23)(18,20,22,24)(25,26,29,30)(27,33,34,31)(28,32,37,38)(35,40,36,39);(1,3,5,4)(2,6,8,7)(11,15)(12,19)(13,21)(16,23)(26,30)(27,34)(32,38)(35,36);(1,2)(3,6)(4,7)(5,8)(9,11)(10,12)(13,18)(14,15)(16,20)(17,19)(21,22)(23,24)(25,27,29,34)(26,31,30,33)(28,36,37,35)(32,40,38,39);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,29,37)(26,32,30,38)(27,35,34,36)(31,39,33,40);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,40)(37,38)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,d^2=f,e^2=fg,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×#49239, #7704953, #7704973, … |
| maximal normal subgroup of | C2×#49239, #7704953, #7704973, … |
| maximal quotient of | #560746, #560935, #561861, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42227);