Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,12)(13,19)(14,27)(15,16)(17,18)(20,28)(21,23)(22,31)(24,33)(25,29)(26,38)(30,32)(34,37)(35,39)(36,40);(1,2,4,6)(3,5,8,7)(10,16)(11,18)(13,25)(20,27)(22,33)(23,30)(26,37)(36,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,17)(12,16)(13,21)(14,22)(15,18)(19,30)(20,31)(23,29)(24,27)(25,32)(26,35)(28,33)(34,39)(36,38)(37,40);(1,3,4,8)(2,5,6,7)(9,12,15,17)(10,18,16,11)(13,23,25,30)(14,24,28,31)(19,32,29,21)(20,33,27,22)(26,36,37,39)(34,40,38,35);(9,13,15,25)(10,19,16,29)(11,21,18,32)(12,23,17,30)(14,26,28,37)(20,34,27,38)(22,35,33,40)(24,36,31,39);(9,14)(10,20)(11,22)(12,24)(13,26)(15,28)(16,27)(17,31)(18,33)(19,34)(21,35)(23,36)(25,37)(29,38)(30,39)(32,40)Polycyclic presentation:
2^8|b^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh70 conjugacy classes — view the full character table.
| maximal subgroup of | #347008, #347018, #347022, … |
| maximal normal subgroup of | #347008, #347018, #347022, … |
| maximal quotient of | #7626375, #7626379, #7626383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55876);