Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #481 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 177 conjugacy classes of subgroups — too many to draw. Only its 61 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:
(2,3)(6,10,11,18)(7,13)(8,16)(12,19,17,15)(14,20);(1,2,4,3)(7,14)(10,18)(13,20)(15,19);(1,3,4,2)(5,6,8,17,9,11,16,12)(7,15,13,10,14,19,20,18)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=dg,e^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^b=eh,d^c=dh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13138, #494878, #494879, … |
| maximal normal subgroup of | C2×#13138, #494878, #494879, … |
| maximal quotient of | #140118, #140144, #140220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6126);