Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2424 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,9,6,4,15,8)(3,10,13,11,14,16,12,7);(1,3,5,13,6,14,15,12)(2,7,9,10,4,11,8,16);(1,4)(2,6)(3,11)(5,9)(7,14)(8,15)(10,13)(12,16);(1,4)(2,6)(3,11)(5,8)(7,14)(9,15)(10,12)(13,16)Polycyclic presentation:
2^8|a^2=f,b^2=f,e^2=h,f^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,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25196, #6438902, #6438903, … |
| maximal normal subgroup of | C2×#25196, #6438902, #6438903, … |
| maximal quotient of | #283579, #283582, #283591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18184);