Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13175 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1943 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1943 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 65 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:
(2,8)(3,12,15,28)(4,6)(5,19)(9,23,25,26)(10,20)(11,17)(13,24,30,32)(14,29,16,27)(18,21);(1,2)(3,13,15,30)(4,17)(5,20)(6,8)(7,11)(9,16,25,14)(10,21)(12,26,28,23)(18,22)(19,31)(24,27,32,29);(1,3,5,12,6,16,19,29,7,15,21,28,4,14,18,27)(2,9,10,24,8,13,22,23,11,25,20,32,17,30,31,26)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,c^b=cg,d^b=dh,e^b=eh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13175, #495047, #495899, … |
| maximal normal subgroup of | C2×#13175, #495047, #495899, … |
| maximal quotient of | #140355, #140370, #140461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6163);