Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,39)(37,38);(1,2,4,6)(3,5,7,8)(10,15)(12,19)(17,21)(20,24)(26,29)(27,34)(32,37)(35,36);(1,3,4,7)(2,5,6,8)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,27)(26,31)(28,36)(29,33)(30,34)(32,40)(35,38)(37,39);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,28,30,38)(26,32,29,37)(27,35,34,36)(31,39,33,40);(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,20,19,24)(25,29,30,26)(27,33,34,31)(28,37,38,32)(35,40,36,39)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=fg,d^2=h,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 | #132491, #253547, #253580, … |
| maximal normal subgroup of | #132491, #253547, #253580, … |
| maximal quotient of | #560390, #560520, #561574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44217);