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) = C210 |
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 | |
2710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(5,6,7,8);(5,6)(7,8)(9,10,12,14)(11,13,16,15);(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15);(9,10)(11,15)(12,14)(13,16)Polycyclic presentation:
2^8|a^2=f,b^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 | #130646, #248460, #248555, … |
| maximal normal subgroup of | #130646, #248460, #248555, … |
| maximal quotient of | #518973, #518974, D4×#385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29201);