Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| 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 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(4,8)(9,10,14,12)(11,16,15,13)(17,18,22,21)(19,24,23,20);(2,6)(3,7)(10,12)(11,15)(17,19)(18,20)(21,24)(22,23);(1,2,5,6)(3,4,7,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,24)(18,23,21,19);(1,3,5,7)(2,4,6,8)(9,12,14,10)(11,16,15,13)(17,18,22,21)(19,20,23,24);(1,4)(2,7)(3,6)(5,8)(9,13,14,16)(10,15,12,11)(17,21,22,18)(19,24,23,20)Polycyclic presentation:
2^8|a^2=f,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #161599, #161883, #162379, … |
| maximal normal subgroup of | #161599, #161883, #162379, … |
| maximal quotient of | #516512, #516540, #516687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28096);