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 | |
692 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)(9,10,13,14)(11,15,19,17)(12,20,21,16)(18,23,24,22)(25,26,30,33)(27,29,32,38)(28,31,35,39)(34,37,36,40);(1,3,5,4)(2,6,8,7)(10,14)(12,21)(15,17)(18,24)(27,32)(28,35)(29,38)(31,39);(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,32)(26,29,33,38)(28,36,35,34)(31,40,39,37);(1,4,5,3)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28)(26,31)(27,34)(29,37)(30,35)(32,36)(33,39)(38,40);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29)(26,32)(27,33)(28,37)(30,38)(31,36)(34,39)(35,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=fh,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×#50263, #7706067, #7706219, … |
| maximal normal subgroup of | C2×#50263, #7706067, #7706219, … |
| maximal quotient of | #561206, #561308, #562402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43251);