Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(2,4)(5,6,9,12,10,13,20,16)(7,14,11,19,17,8,18,15);(5,7,15,6,10,17,19,13)(8,16,20,18,14,12,9,11);(1,2,3,4)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=e,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dgh,f^b=fgh,g^b=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132205, #132207, #132209, … |
| maximal normal subgroup of | #132205, #132207, #132209, … |
| maximal quotient of | #140612, #140628, #140654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6214);