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 | |
697 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)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,26,29,30)(27,31,33,37)(28,32,36,38)(34,39,35,40);(1,3,5,4)(2,6,8,7)(10,15)(12,16)(18,21)(20,23)(26,30)(27,33)(32,38)(34,35);(1,2)(3,6)(4,7)(5,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27)(26,31)(28,35)(29,33)(30,37)(32,40)(34,36)(38,39);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,29,36)(26,32,30,38)(27,34,33,35)(31,39,37,40);(1,4,5,3)(2,7,8,6)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,40)(36,38)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fh,d^2=f,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×#50412, #7705817, #7706227, … |
| maximal normal subgroup of | C2×#50412, #7705817, #7706227, … |
| maximal quotient of | #560757, #560992, #561850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43400);