Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | Aut(G) = C23×#7737411 (order 4096) · Inn(G) = #390 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,9,5,8,4,14)(3,10,13,15,11,16,12,7)(17,18,22,26,23,25,21,19)(20,28,30,32,31,27,29,24);(1,3)(2,7,8,15)(4,12)(5,11)(6,13)(9,10,14,16)(17,19,22,25,23,26,21,18)(20,28,30,24,31,27,29,32);(2,8)(3,11)(9,14)(12,13)(17,20,22,30,23,31,21,29)(18,24,19,27,25,32,26,28)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eg,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,f^b=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9766, #205586, #205587, … |
| maximal normal subgroup of | C2×#9766, #205586, #205587, … |
| maximal quotient of | #102038, #102039, #102040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2754);