Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,15)(11,12,16,14)(17,18,20,22)(19,23,24,21);(1,2)(3,7)(4,5)(6,8)(9,11,13,16)(10,14,15,12);(2,5)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,20,24)(18,21,22,23);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6688185, #6688197, #6688231, … |
| maximal normal subgroup of | #6688185, #6688197, #6688231, … |
| maximal quotient of | #544561, #545104, #545559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35477);