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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(6,8)(10,11)(15,16);(2,4)(9,10,14,11)(12,15,13,16);(1,2)(3,4)(5,6)(7,8)(9,11,14,10)(12,16,13,15);(2,4)(9,12,14,13)(10,15,11,16);(1,2)(3,4)(9,13,14,12)(10,16,11,15)Polycyclic presentation:
2^8|b^2=f,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 | #130742, #248479, #248480, … |
| maximal normal subgroup of | #130742, #248479, #248480, … |
| maximal quotient of | #519006, #519030, #519132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29091);