Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3061 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
204 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,11)(8,12)(9,14)(10,16)(13,21)(15,22)(17,30)(18,27)(19,25)(20,26)(23,29)(24,31)(28,34)(32,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,18)(9,17)(10,20)(12,27)(13,24)(14,30)(15,25)(16,26)(19,22)(21,31)(23,32)(28,35)(29,36)(33,34);(1,3,4,2)(6,12)(7,17)(8,21)(9,22)(11,24)(13,15)(14,28)(16,29)(18,32)(19,20)(23,34)(25,36)(26,35)(27,30)Polycyclic presentation:
2^8|c^2=d,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,f^b=fh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13384, #499010, #499011, … |
| maximal normal subgroup of | C2×#13384, #499010, #499011, … |
| maximal quotient of | #144682, #144685, #144786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6372);