Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
165 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(5,6,13,17,20,35,9,16,12,19,10,26,11,18,25,29)(7,15,22,32,28,30,21,31,24,34,8,14,23,33,27,36);(1,2)(3,4)(5,7,20,23,12,24,11,28)(6,14,18,36,19,32,35,31)(8,13,21,9,22,10,27,25)(15,29,30,26,34,16,33,17);(2,4)(5,8)(6,15)(7,13)(9,23)(10,24)(11,27)(12,22)(14,29)(16,32)(17,31)(18,30)(19,34)(20,21)(25,28)(26,36)(33,35)Polycyclic presentation:
2^8|a^2=dg,b^2=degh,d^2=g,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=eg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13534, #499540, #499541, … |
| maximal normal subgroup of | C2×#13534, #499540, #499541, … |
| maximal quotient of | #146197, #146198, #146205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6522);