Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
652 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)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(1,2,4,6)(3,5,7,8)(10,14)(12,21)(15,17)(18,24)(26,31)(28,32)(33,37)(36,39);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #98479, #260744, #260746, … |
| maximal normal subgroup of | #98479, #260744, #260746, … |
| maximal quotient of | #557081, #557115, #557365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39148);