Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(1,2,3,4)(5,6)(7,16)(8,21)(9,11)(10,15)(12,20)(13,23)(14,22)(17,25)(18,32)(19,28)(24,26)(27,31)(29,36)(30,33)(34,35);(6,11,15,14)(7,17,20,19)(13,26,29,21)(18,31,34,33);(2,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,25)(13,27)(14,28)(15,16)(17,22)(21,35)(23,33)(24,34)(26,32)(29,30)(31,36);(5,8,22,36,10,24,9,23)(6,13,11,26,15,29,14,21)(7,18,17,31,20,34,19,33)(12,27,25,32,16,30,28,35)Polycyclic presentation:
2^8|a^2=fh,b^2=egh,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #143197, #143199, #143201, … |
| maximal normal subgroup of | #143197, #143199, #143201, … |
| maximal quotient of | #321905, #321909, #321913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21493);