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 | |
385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,7,8)(9,10,12,11)(14,19)(15,20)(16,17);(1,2)(3,4)(9,11,12,10)(13,14)(15,16)(17,19)(18,20);(2,4)(10,11)(13,15,17,20,18,19,16,14)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=dh,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11316, #358354, #358355, … |
| maximal normal subgroup of | C2×#11316, #358354, #358355, … |
| maximal quotient of | #121084, #121108, #121112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4304);