Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12314 |
| automorphism | Aut(G) = #72279473 (order 2048) · Inn(G) = #1619 · Out(G) = C22⋊C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
192 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 39 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 — 3 generators on 16 points:
(2,6)(3,10)(4,14)(7,15)(8,16)(9,12)(11,13);(3,11)(4,14)(6,15)(7,9)(10,13)(12,16);(1,2)(3,12)(4,7)(5,8)(6,10)(9,14)(11,16)(13,15)Polycyclic presentation:
2^8|d^2=f,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,g^a=gh,d^b=df,e^b=eg,g^b=gh,d^c=dg,e^c=egh,f^c=fh,g^c=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | #153625, #153628, #153630, … |
| maximal normal subgroup of | #153625, #153628, #153630, … |
| maximal quotient of | #123331, #123332, #123333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5302);