Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
207 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,11,9,12,7,10)(13,14,17,19,16,18,20,15);(1,2,3,4)(6,10)(7,8)(11,12)(14,18)(15,19);(5,7,9,8)(6,10,12,11)(13,15,20,18,16,19,17,14)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,b^a=bd,c^b=ch,d^b=dg,e^b=eg88 conjugacy classes — view the full character table.
| maximal subgroup of | #92935, #92946, #92951, … |
| maximal normal subgroup of | #92935, #92946, #92951, … |
| maximal quotient of | #96888, #96891, #96904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1270);