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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
771 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)(9,10)(11,13)(12,15)(14,16)(19,23)(21,24);(2,4)(3,5)(9,11,14,12)(10,15,16,13)(17,18,20,22)(19,21,23,24);(1,3,6,5)(2,7,4,8)(9,12,14,11)(10,15,16,13)(17,19)(18,21)(20,23)(22,24);(1,4,6,2)(3,7,5,8)(9,13,14,15)(10,11,16,12);(1,5,6,3)(2,7,4,8)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=f,d^2=f,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 | #7040292, #7040296, #7040307, … |
| maximal normal subgroup of | #7040292, #7040296, #7040307, … |
| maximal quotient of | #535326, #535539, #535957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38317);