Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
976 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 976 conjugacy classes of subgroups — too many to draw. Only its 141 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 — 4 generators on 16 points:
(1,2)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15);(6,15)(9,16);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14);(4,14)(7,12)(9,16)(11,13)Polycyclic presentation:
2^8|f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139742, #144364, #144382, … |
| maximal normal subgroup of | #139742, #144364, #144382, … |
| maximal quotient of | #245063, #245066, #245071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16885);