Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | Aut(G) = #72340198 (order 4096) · Inn(G) = #2631 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2631 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2,5,4,6,9,15,8)(3,10,12,16,14,11,13,7);(1,3)(2,7)(4,11)(5,13)(6,14)(8,10)(9,16)(12,15);(2,8)(4,9)(5,15)(7,11)(10,16)(12,13);(1,4,15,2,6,8,5,9)(3,11,12,7,14,10,13,16)Polycyclic presentation:
2^8|a^2=e,d^2=fh,e^2=h,f^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,d^c=de,e^c=eh,f^c=fh,g^d=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #143162, #143164, #143166, … |
| maximal normal subgroup of | #143162, #143164, #143166, … |
| maximal quotient of | #331008, #331011, #331012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23045);