Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | #2540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,4)(3,7,5,8)(10,13)(14,16)(17,18,22,21)(19,24,20,23);(2,4)(3,5)(17,19,22,20)(18,23,21,24);(1,3,6,5)(2,7,4,8)(9,10,12,13)(11,14,15,16)(17,20,22,19)(18,23,21,24);(1,4,6,2)(3,7,5,8)(9,11,12,15)(10,14,13,16)(17,21,22,18)(19,23,20,24);(1,5,6,3)(2,7,4,8)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=fg,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7010362, #7010365, #7010377, … |
| maximal normal subgroup of | #7010362, #7010365, #7010377, … |
| maximal quotient of | #535549, #535551, #536190, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38290);