Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
853 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 853 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,14)(7,16,12,19)(8,17,18,11)(9,15,20,13);(5,7,10,12)(6,8,14,18)(9,17,20,11)(13,19,15,16);(2,4)(5,8)(6,11)(7,15)(9,16)(10,18)(12,13)(14,17)(19,20);(5,8)(6,12)(7,14)(9,19)(10,18)(11,13)(15,17)(16,20)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151146, #151148, #151373, … |
| maximal normal subgroup of | #151146, #151148, #151373, … |
| maximal quotient of | #244501, #244508, #244512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16878);