Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(1,2)(3,4)(5,6,10,14,11,8,22,15)(7,16,20,31,21,33,36,32)(9,24,17,13,25,29,34,23)(12,26,27,30,18,35,28,19);(5,7,17,26,10,20,25,30,11,21,34,35,22,36,9,19)(6,12,24,32,15,28,23,33,8,18,29,31,14,27,13,16);(2,4)(5,8)(6,11)(7,18)(9,23)(10,14)(12,21)(13,25)(15,22)(16,30)(17,29)(19,33)(20,27)(24,34)(26,31)(28,36)(32,35)Polycyclic presentation:
2^8|a^2=f,b^2=dg,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152174, #152175, #152297, … |
| maximal normal subgroup of | #152174, #152175, #152297, … |
| maximal quotient of | #134677, #134698, #134769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5830);