Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,17)(11,15,18,13)(12,16,22,24)(19,23,20,21)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,40)(37,39);(1,2,4,6)(3,5,7,8)(11,18)(13,15)(19,20)(21,23)(26,31)(29,39)(32,40)(33,36);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,17)(14,18)(16,21)(19,22)(23,24)(25,27)(26,32)(28,37)(29,33)(30,35)(31,40)(34,38)(36,39);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,28,30,38)(26,33,31,36)(27,34,35,37)(29,32,39,40);(9,13,14,15)(10,11,17,18)(12,21,22,23)(16,19,24,20)(25,29,30,39)(26,34,31,37)(27,33,35,36)(28,32,38,40)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6679239, #6679242, #6679255, … |
| maximal normal subgroup of | #6679239, #6679242, #6679255, … |
| maximal quotient of | #544448, #545034, #545522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35645);