Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = C24×#3109 (order 2048) · Inn(G) = #442 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,14,9,15,18,13,10,16,20,11,19,12,7,17);(1,2,3,4)(6,11)(7,8)(9,19)(12,17)(13,15)(14,16)(18,20);(6,12,16,15)(7,18)(8,20)(11,17,14,13)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=de,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=dh,d^b=dg,e^b=egh,g^b=gh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12677, #493158, #493159, … |
| maximal normal subgroup of | C2×#12677, #493158, #493159, … |
| maximal quotient of | #129120, #129124, #129132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5665);