Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| 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 | #3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,9,13,10,8,17,12)(7,11,15,19,16,14,20,18);(2,3)(5,7,9,15,10,16,17,20)(6,11,13,19,8,14,12,18);(5,8)(6,10)(7,14)(9,12)(11,16)(13,17)(15,18)(19,20);(1,3,4,2)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,f^c=fh70 conjugacy classes — view the full character table.
| maximal subgroup of | #228174, #228176, #228182, … |
| maximal normal subgroup of | #228174, #228176, #228182, … |
| maximal quotient of | #229054, #229055, #229056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16003);