Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2410 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,9,6,10,5,8)(3,11,14,16,15,12,13,7);(1,3)(2,7)(4,14)(5,13)(6,15)(8,12)(9,11)(10,16);(1,2)(3,12)(4,8)(5,9)(6,10)(7,13)(11,15)(14,16);(1,2)(3,7)(4,9)(5,8)(6,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^8|a^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24472, #6437949, #6437950, … |
| maximal normal subgroup of | C2×#24472, #6437949, #6437950, … |
| maximal quotient of | #268283, #268284, #268285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17460);