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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
219 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 219 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:
(1,2)(3,4)(5,6,10,14,12,15,11,18)(7,9,16,20,17,8,13,19);(2,4)(5,7,9,14,11,16,19,15,12,17,8,18,10,13,20,6);(5,8,10,19,12,9,11,20)(6,13,14,7,15,16,18,17)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=g,e^2=h,f^2=g,g^2=h,b^a=bd,d^a=dg,f^a=fg,g^a=gh,c^b=cg,e^b=eh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12650, #493084, #493085, … |
| maximal normal subgroup of | C2×#12650, #493084, #493085, … |
| maximal quotient of | #128557, #128574, #128608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5638);