Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9781 |
| automorphism | Aut(G) order 32768 |
| 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 | C2×C42 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
305 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 79 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)(9,10,12,15,13,14,16,11);(1,2,5,7,6,8,4,3)(9,11,12,14,13,15,16,10);(1,3,4,8,6,7,5,2)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=egh,d^2=gh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138858, #138859, #138870, … |
| maximal normal subgroup of | #138858, #138859, #138870, … |
| maximal quotient of | #102068, #102084, #102175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2790);