Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | Aut(G) = C22×#72232559 (order 4096) · Inn(G) = #390 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
276 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,3)(4,5)(7,8)(9,10)(11,13,15,16)(12,14);(1,2,4,3,6,8,5,7)(9,11)(10,13,14,16)(12,15);(1,3,4,8,6,7,5,2)(10,14)(11,15)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eg,d^2=gh,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cg,d^b=dh,f^b=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9786, #205746, #205749, … |
| maximal normal subgroup of | C2×#9786, #205746, #205749, … |
| maximal quotient of | #102071, #102072, #102075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2774);