Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
258 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 105 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:
(1,2,4,3)(6,10)(7,8)(11,12)(13,14)(15,19)(16,18)(17,20);(5,6,8,12,9,11,7,10)(14,18)(15,16)(19,20);(1,3,4,2)(6,11)(10,12)(13,15,17,16)(14,19,20,18)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eh,d^2=g,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97281, #97282, #97286, … |
| maximal normal subgroup of | #97281, #97282, #97286, … |
| maximal quotient of | #96729, #96740, #96748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1221);