Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | Aut(G) = C22×#7622880 (order 2048) · Inn(G) = #535 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2404 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 625 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(6,14)(9,16);(1,2)(3,9,10,16)(4,8)(5,7)(6,11,14,12)(13,15);(1,3)(2,6)(4,12)(5,10)(7,14)(8,16)(9,15)(11,13);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^8|b^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fh,e^c=eh,g^c=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #152686, #152687, #152688, … |
| maximal normal subgroup of | #152686, #152687, #152688, … |
| maximal quotient of | #251791, #251792, #251793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16982);