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 | #3552 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(17,18,22,26)(19,23,28,36)(20,30,31,45)(21,33,34,25)(24,38,39,32)(27,40,41,48)(29,44,43,37)(35,46,47,42);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,28)(18,23,26,36)(20,27,31,41)(21,29,34,43)(24,35,39,47)(25,37,33,44)(30,40,45,48)(32,42,38,46);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,32,22,31,34,38)(18,24,25,30,26,39,33,45)(19,27,29,42,28,41,43,46)(23,35,37,40,36,47,44,48);(3,12)(7,10)(11,13)(15,16)(20,31)(24,39)(27,41)(30,45)(32,38)(35,47)(40,48)(42,46);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,28)(20,31)(23,36)(24,39)(29,43)(30,45)(32,38)(37,44)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fg,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,d^c=dh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6367648, #6367651, #6367657, … |
| maximal normal subgroup of | #6367648, #6367651, #6367657, … |
| maximal quotient of | #6386969, #6389674, #6389693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55599);