Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C2×D4 · Out(G) = C210 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3)(9,10,11,12)(13,14,16,18,17,19,20,15);(1,2,4,3)(5,6)(7,8)(14,15)(16,20)(18,19);(1,3,4,2)(6,8)(13,15,16,14,17,18,20,19)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=d,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11293, #358314, #358315, … |
| maximal normal subgroup of | C2×#11293, #358314, #358315, … |
| maximal quotient of | #120697, #120698, #120744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4281);