Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
922 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)(11,12)(13,16)(14,15)(18,19)(22,23);(1,2)(3,7)(4,8)(5,6)(10,15)(11,13)(17,18,21,19)(20,22,24,23);(2,6)(3,8)(9,11,14,13)(10,12,15,16)(17,19,21,18)(20,23,24,22);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,15)(14,16)(17,20,21,24)(18,22,19,23);(1,4,5,7)(2,3,6,8)(9,13,14,11)(10,16,15,12)Polycyclic presentation:
2^8|b^2=g,c^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #6580787, #6580790, #6580850, … |
| maximal normal subgroup of | #6580787, #6580790, #6580850, … |
| maximal quotient of | #519021, #519045, #519079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29440);