Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) = #72257321 (order 1024) · Inn(G) = #386 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1869 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 81 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:
(1,2,7,12,6,11,25,29,8,13,27,31,5,24,15,16)(3,14,18,32,19,4,21,23,20,22,10,30,17,9,28,26);(1,3,15,28,5,17,27,10,8,20,25,21,6,19,7,18)(2,9,29,32,24,22,12,26,13,4,16,30,11,14,31,23);(1,4)(2,10)(3,16)(5,14)(6,22)(7,26)(8,9)(11,28)(12,19)(13,18)(15,30)(17,31)(20,29)(21,24)(23,27)(25,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^b=eh,d^c=dh,e^c=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12986, #494428, #494957, … |
| maximal normal subgroup of | C2×#12986, #494428, #494957, … |
| maximal quotient of | #136579, #136587, #136595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5974);