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 | |
706 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)(10,14)(15,17)(16,21)(22,23)(25,26,30,28)(27,33,35,31)(29,36,38,32)(34,39,37,40);(10,14)(12,20)(15,17)(18,24)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(1,3,5,8)(2,6,4,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,4,5,2)(3,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,30,26)(27,31,35,33)(29,36,38,32)(34,39,37,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^8|a^2=fgh,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×#46943, #7700776, #7700792, … |
| maximal normal subgroup of | C2×#46943, #7700776, #7700792, … |
| maximal quotient of | #576838, #576882, #577077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39931);