Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,17)(8,20,22,28)(9,23)(10,12)(11,16)(13,18,31,36)(14,25)(15,19)(21,33,24,34)(26,32,35,30)(27,29);(1,2,4,3)(5,7,9,10,11,19,25,27)(6,12,14,17,16,29,23,15)(8,18,24,26,22,36,21,35)(13,28,32,34,31,20,30,33);(1,3,4,2)(5,8,11,22)(6,13,16,31)(7,18,19,36)(9,24,25,21)(10,26,27,35)(12,28,29,20)(14,32,23,30)(15,33,17,34)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=dg,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,e^b=eh,d^c=dh,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13141, #494886, #494887, … |
| maximal normal subgroup of | C2×#13141, #494886, #494887, … |
| maximal quotient of | #140120, #140146, #140220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6129);