Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3562 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,35)(28,41)(29,38)(30,39)(31,45)(33,40)(34,46)(37,43)(42,47)(44,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,37)(26,38)(27,39)(32,41)(33,42)(34,44)(35,43)(36,45)(40,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,35,23,34,21,33)(18,25,26,32,27,40,36,46)(19,28,31,43,30,44,29,42)(24,37,38,41,39,47,45,48);(3,12)(7,10)(11,13)(15,16)(17,21,23,22)(18,26,27,36)(19,29,30,31)(20,33,34,35)(24,38,39,45)(25,32,40,46)(28,42,44,43)(37,41,47,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,30)(20,34)(24,39)(25,40)(29,31)(32,46)(33,35)(38,45)Polycyclic presentation:
2^8|c^2=fg,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62600, #10580027, #10580055, … |
| maximal normal subgroup of | C2×#62600, #10580027, #10580055, … |
| maximal quotient of | #6386967, #6389672, #6389690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55588);