Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(5,6,8,7)(10,13)(14,16);(5,7,8,6)(9,10)(11,14)(12,13)(15,16);(2,4);(10,13)(14,16);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=f,b^a=bf,c^a=cg,d^b=df100 conjugacy classes — view the full character table.
| maximal subgroup of | #106181, #356646, #356647, … |
| maximal normal subgroup of | #106181, #356646, #356647, … |
| maximal quotient of | C22⋊C4×#117, #513485, #513494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27124);