Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | Aut(G) = #72338516 (order 4096) · Inn(G) = #2684 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2684 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 402 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:
(2,7)(3,11)(4,5)(8,12)(9,10)(13,14);(1,2,5,9,6,10,4,7)(3,12,11,16,13,8,14,15);(1,3)(2,8)(4,11)(5,14)(6,13)(7,15)(9,16)(10,12);(2,9)(4,5)(7,10)(8,16)(11,14)(12,15)Polycyclic presentation:
2^8|b^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153189, #153190, #153193, … |
| maximal normal subgroup of | #153189, #153190, #153193, … |
| maximal quotient of | #345808, #345810, #345812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24270);