Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62295 |
| 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 | #3251 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,27)(21,30)(22,25)(24,28)(26,32)(29,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,27,32,31);(17,20,21,28,22,29,19,26)(18,24,23,27,25,32,30,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,31,32,27)Polycyclic presentation:
2^8|b^2=fh,c^2=g,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6363136, #6363141, #6363172, … |
| maximal normal subgroup of | #6363136, #6363141, #6363172, … |
| maximal quotient of | #6386847, #6389340, #6389343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55384);