Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,9)(7,11,12,10)(13,14,17,15,19,16,18,20);(1,2)(3,4)(6,9)(10,11)(13,15)(14,17)(16,18)(19,20);(2,4)(13,14,19,16)(15,17,20,18);(5,7,8,12)(6,10,9,11)(13,16,19,14)(15,18,20,17)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=efh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,e^b=eh,f^b=fh,e^c=eh,f^c=fh,e^d=eh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26907, #6439657, #6439659, … |
| maximal normal subgroup of | C2×#26907, #6439657, #6439659, … |
| maximal quotient of | #312177, #312179, #312194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19895);