Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 167 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 — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16,12,17,10,15)(7,18,22,13,23,30,21,31)(8,24,25,33,26,32,9,14)(19,35,27,34,20,28,36,29);(1,3,4,2)(5,7)(6,13)(8,19)(9,27)(10,22)(11,21)(12,23)(14,28)(15,30)(16,18)(17,31)(20,26)(24,34)(25,36)(29,32)(33,35);(5,8,11,25,12,26,10,9)(6,14,15,32,17,33,16,24)(7,19,21,36,23,20,22,27)(13,28,30,29,31,35,18,34);(5,9,10,26,12,25,11,8)(6,14,15,32,17,33,16,24)(7,20,21,27,23,19,22,36)(13,29,18,28,31,34,30,35)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=fh,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #498316, #498321, #498340, … |
| maximal normal subgroup of | #498316, #498321, #498340, … |
| maximal quotient of | #231467, #231472, #231476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16187);