Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,6,8,7)(9,10,13,15)(11,12,14,16)(18,19)(23,24);(2,4)(6,7)(9,11)(10,12)(13,14)(15,16)(17,18,22,19)(20,23,21,24);(1,2,5,4)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,19,22,18)(20,24,21,23);(1,3,5,8)(2,6,4,7)(9,11)(10,12)(13,14)(15,16)(17,20,22,21)(18,23,19,24);(1,4,5,2)(3,6,8,7)(9,12)(10,14)(11,15)(13,16)(17,21,22,20)(18,24,19,23)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=f,d^2=f,e^2=f,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 | #98201, #223012, #223040, … |
| maximal normal subgroup of | #98201, #223012, #223040, … |
| maximal quotient of | #519008, #519066, #519123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29164);