Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
708 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)(10,14)(15,17)(16,21)(22,23)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,3,5,4)(2,6,8,7)(10,14)(12,20)(15,17)(18,24)(26,28)(27,34)(29,37)(38,39);(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,3,5,4)(2,6,8,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,4,5,3)(2,7,8,6)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=gh,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47449, #7701341, #7701402, … |
| maximal normal subgroup of | C2×#47449, #7701341, #7701402, … |
| maximal quotient of | #576828, #576885, #577079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40437);