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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26,30,31)(27,28,32,36)(29,37,38,33)(34,40,39,35);(1,2,5,4)(3,6,8,7)(10,14)(15,21)(16,19)(22,23)(26,31)(27,32)(29,38)(35,40);(1,3,5,8)(2,6,4,7)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,40)(38,39);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,37)(38,40);(9,12,13,20)(10,16,14,19)(11,17,18,24)(15,22,21,23)(25,29,30,38)(26,33,31,37)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fgh,d^2=g,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×#50334, #7705974, #7706221, … |
| maximal normal subgroup of | C2×#50334, #7705974, #7706221, … |
| maximal quotient of | #561188, #561296, #562276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43322);