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 | |
677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,12)(11,17,18,15)(13,16,21,20)(19,23,24,22)(26,30)(28,36)(31,33)(34,40);(11,18)(13,21)(15,17)(16,20)(25,26,29,30)(27,33,35,31)(28,37,36,32)(34,38,40,39);(1,2,4,5)(3,6,7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=fg,d^2=g,e^2=f,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×#46920, #7700737, #7700792, … |
| maximal normal subgroup of | C2×#46920, #7700737, #7700792, … |
| maximal quotient of | #576806, #576864, #577121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39908);