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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
679 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)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,38)(37,39);(1,3,5,4)(2,6,8,7)(10,12)(15,17)(16,20)(22,23)(26,29)(27,34)(28,36)(39,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27)(26,31)(28,37)(29,33)(30,34)(32,40)(35,36)(38,39);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,36)(26,32,29,38)(27,35,34,37)(31,39,33,40);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,30,26)(27,33,34,31)(28,38,36,32)(35,40,37,39)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,d^2=f,e^2=g,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×#46989, #7700704, #7700737, … |
| maximal normal subgroup of | C2×#46989, #7700704, #7700737, … |
| maximal quotient of | #576862, #576878, #577122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39977);