Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2420 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,15)(5,17)(9,24)(10,20)(13,14)(16,28)(18,22)(21,29)(25,26)(31,32);(1,2,5,7,6,10,17,20)(3,8,11,22,14,23,13,18)(4,9,15,19,16,24,28,30)(12,21,25,31,27,32,26,29);(1,3)(2,8)(4,12)(5,13)(6,14)(7,18)(9,21)(10,23)(11,17)(15,26)(16,27)(19,29)(20,22)(24,32)(25,28)(30,31);(1,4,5,15,6,16,17,28)(2,9,7,19,10,24,20,30)(3,12,13,26,14,27,11,25)(8,21,18,29,23,32,22,31)Polycyclic presentation:
2^8|b^2=f,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24987, #6447051, #6447052, … |
| maximal normal subgroup of | C2×#24987, #6447051, #6447052, … |
| maximal quotient of | #280843, #280846, #280849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17975);