Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | Aut(G) = C4×#23882 (order 1024) · Inn(G) = #418 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #418 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | #1490 |
| prime radical | #575 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,5,9,6,12,4,20,7,13,23,10,22,11,24,28)(3,14,15,27,18,29,21,30,19,31,16,8,17,25,32,26);(1,3)(2,8)(4,21)(5,15)(6,18)(7,19)(9,25)(10,29)(11,30)(12,26)(13,27)(14,20)(16,23)(17,22)(24,32)(28,31);(1,4,6,23,7,24,22,5)(2,9,11,28,13,10,12,20)(3,15,17,32,19,16,18,21)(8,14,26,29,27,31,30,25)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=fh,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cfh,d^b=df,f^b=fh,d^c=dh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139696, #139697, #139698, … |
| maximal normal subgroup of | #139696, #139697, #139698, … |
| maximal quotient of | #124649, #124669, #124678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4658);