Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C2×#72237528 (order 2048) · Inn(G) = #120 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #120 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1869 |
| prime radical | #532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
237 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 237 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:
(1,2,6,12,7,13,25,31,8,11,26,30,5,10,15,28)(3,14,18,23,16,29,32,4,17,20,24,19,22,9,27,21);(1,3,15,32,5,22,26,18,8,17,25,27,7,16,6,24)(2,9,28,23,10,20,30,21,11,29,31,19,13,14,12,4);(1,4)(2,3)(5,23)(6,14)(7,19)(8,21)(9,15)(10,22)(11,17)(12,24)(13,16)(18,30)(20,26)(25,29)(27,31)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,e^a=eh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #94136, #94137, #94138, … |
| maximal normal subgroup of | #94136, #94137, #94138, … |
| maximal quotient of | #136576, #136584, #136592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5970);