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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
688 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)(10,14)(15,18)(16,21)(22,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,3,5,2)(4,7,8,6)(10,14)(11,17)(12,20)(22,24)(26,30)(31,37)(32,35)(38,39);(1,4,5,8)(2,6,3,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,28,29,36)(26,32,30,35)(27,33,34,40)(31,38,37,39)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=fg,d^2=h,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×#51238, #7707269, #7707271, … |
| maximal normal subgroup of | C2×#51238, #7707269, #7707271, … |
| maximal quotient of | #560359, #560554, #561541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44226);