Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,14,11,15,9,19)(7,16,17,12,8,13,18,20);(5,7,10,17,11,8,9,18)(6,12,14,13,15,20,19,16);(2,4);(5,8)(6,13)(7,11)(9,17)(10,18)(12,19)(14,20)(15,16)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=deh,e^b=eh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22888, #6381454, #6381456, … |
| maximal normal subgroup of | C2×#22888, #6381454, #6381456, … |
| maximal quotient of | #227142, #227143, #227144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15876);