Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,8)(3,7,4,5)(10,14)(12,15)(18,21)(22,24);(3,4)(5,7)(9,10,13,14)(11,15,16,12);(1,3,6,4)(2,5,8,7)(9,11)(10,12)(13,16)(14,15)(17,18,20,21)(19,22,23,24);(1,4,6,3)(2,7,8,5)(9,12)(10,11)(13,15)(14,16);(1,5,6,7)(2,3,8,4)(9,10)(11,12)(13,14)(15,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=g,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97183, #172565, #172566, … |
| maximal normal subgroup of | #97183, #172565, #172566, … |
| maximal quotient of | #557222, #557540, #557663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39188);