Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) = #72252039 (order 2048) · Inn(G) = D42 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2358 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 165 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:
(1,2,6,11,8,12,7,10)(3,13,15,16,5,4,14,9);(1,3,6,15,8,5,7,14)(2,9,11,13,12,16,10,4);(1,4)(2,3)(5,12)(6,9)(7,16)(8,13)(10,14)(11,15);(1,5)(2,4)(3,8)(6,14)(7,15)(9,11)(10,16)(12,13)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,d^c=dh,f^c=fh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #152458, #152467, #152556, … |
| maximal normal subgroup of | #152458, #152467, #152556, … |
| maximal quotient of | #226938, #226939, #226940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15887);