Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,19,26,34)(11,15,22,28,20,16,29,31)(12,23,24,37,25,17,32,33)(21,30,36,40,35,27,38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,19)(17,27)(18,28)(23,30)(24,36)(25,35)(26,29)(31,34)(32,38)(33,39)(37,40);(3,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,30)(18,31)(23,27)(24,36)(25,35)(26,29)(28,34)(32,38)(33,40)(37,39);(2,6)(5,7)(11,20)(15,16)(21,35)(22,29)(27,30)(28,31)(36,38)(39,40);(9,12,13,24,14,25,26,32)(10,17,18,33,19,23,34,37)(11,21,22,36,20,35,29,38)(15,27,28,39,16,30,31,40)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,e^a=eh,c^b=cf,d^b=dh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129592, #129595, #129655, … |
| maximal normal subgroup of | #129592, #129595, #129655, … |
| maximal quotient of | #6370406, #6370471, #6370472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53306);