Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8 | C22 |
| commutator | C2 | C2×C4×C16 |
| Frattini, cosocle | C22×C8 | C23 |
| socle, prime radical | C23 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(1,2,3,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=d,e^2=g,g^2=h,b^a=bd160 conjugacy classes — view the full character table.
| maximal subgroup of | #93249, #93251, #93308, … |
| maximal normal subgroup of | #93249, #93251, #93308, … |
| maximal quotient of | #128203, #128209, #128214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5578);