Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | Aut(G) = #72305983 (order 2048) · Inn(G) = #3109 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(2,6)(3,10)(4,13)(7,12)(8,14)(9,15)(11,16);(3,11)(6,14)(10,16)(12,15);(1,2)(3,12)(4,7)(5,8)(6,10)(9,13)(11,15)(14,16);(2,7,8,9)(3,11)(6,12,14,15)(10,16)Polycyclic presentation:
2^8|d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #143162, #143163, #143166, … |
| maximal normal subgroup of | #143162, #143163, #143166, … |
| maximal quotient of | #494052, #494997, #495481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26676);