Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | Aut(G) order 16384 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
531 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 531 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,14)(8,10)(9,13)(11,15)(12,17)(16,18)(19,20);(6,10,13,12)(11,18,19,14);(2,4)(5,6)(7,11)(8,12)(9,13)(10,17)(14,15)(16,19)(18,20);(5,7,8,15,9,16,17,20)(6,11,12,14,13,19,10,18)Polycyclic presentation:
2^8|a^2=f,b^2=egh,d^2=e,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 | #143202, #143203, #143204, … |
| maximal normal subgroup of | #143202, #143203, #143204, … |
| maximal quotient of | #321902, #321907, #321910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21490);