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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(1,3,5,2)(4,7,8,6)(11,17)(13,21)(15,19)(16,20)(26,31)(27,33)(32,38)(34,39);(1,4)(2,6)(3,7)(5,8)(9,11,14,17)(10,15,12,19)(13,18,21,23)(16,22,20,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fh,d^2=fh,e^2=f,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 | #132440, #253675, #253677, … |
| maximal normal subgroup of | #132440, #253675, #253677, … |
| maximal quotient of | #544464, #545039, #545534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35386);