Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
4747 conjugacy classes of subgroups; only the 2859 normal subgroups are shown.
This group has 4747 conjugacy classes of subgroups — too many to draw. Only its 2859 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,13)(12,15)(14,16);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,16)(13,15);(2,7)(3,10)(6,13)(8,15)(9,14)(11,16);(2,8)(6,9)(7,15)(13,14);(3,11)(6,14)(9,13)(10,16);(2,7)(6,14)(8,15)(9,13)Polycyclic presentation:
2^8|b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55805);