Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#72256923 (order 2048) · Inn(G) = C22 · Out(G) = C2×#23794 |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C16 | C22 |
| commutator | C2 | C4×C32 |
| socle, prime radical | C23 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,4)(9,10,11,15,12,16,19,25,13,17,20,26,22,28,31,35,14,18,21,27,23,29,32,36,24,30,33,37,34,38,39,40);(2,4)(5,6,7,8)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,b^a=bc160 conjugacy classes — view the full character table.
| maximal subgroup of | #93886, #93890, #93915, … |
| maximal normal subgroup of | #93886, #93890, #93915, … |
| maximal quotient of | #93989, #93998, #94005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 453);