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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
841 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(17,18,21,20)(19,23,24,22);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,21,24)(18,22,20,23);(2,5)(6,7)(10,14)(11,16)(17,18,21,20)(19,22,24,23);(1,2)(3,7)(4,5)(6,8)(9,11,13,16)(10,15,14,12)(17,20,21,18)(19,23,24,22);(1,3,4,8)(2,6,5,7)(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=fgh,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 | #98174, #164356, #164435, … |
| maximal normal subgroup of | #98174, #164356, #164435, … |
| maximal quotient of | #519114, #519116, #519242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29485);