Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12301 |
| automorphism | Aut(G) = C2×#6346637 (order 1024) · Inn(G) = #1613 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3204 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
161 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,21,6,11,22,10)(3,12,15,27,16,26,14,8)(4,17,19,9,20,7,18,24)(13,25,31,29,32,28,30,23);(2,7)(4,18)(5,22)(8,23)(9,21)(10,24)(11,17)(12,28)(13,30)(14,15)(19,20)(25,26)(27,29)(31,32);(1,3,6,16)(2,8,11,27)(4,13,20,32)(5,15,22,14)(7,23,17,29)(9,28,24,25)(10,26,21,12)(18,30,19,31)Polycyclic presentation:
2^8|a^2=fg,c^2=h,d^2=f,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gh,d^b=dfh,e^b=egh,f^b=fh,d^c=dg,e^c=eh,e^d=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12303, #484575, #484576, … |
| maximal normal subgroup of | C2×#12303, #484575, #484576, … |
| maximal quotient of | #126284, #126439, #126442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5291);