Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
759 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 759 conjugacy classes of subgroups — too many to draw. Only its 343 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,10)(8,15)(11,16)(14,19);(5,6,8,12,9,10,15,17)(7,11,14,18,13,16,19,20);(1,3,4,2)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20);(7,13)(11,16)(14,19)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,g^2=h,b^a=be,g^a=gh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| maximal subgroup of | #147325, #147326, #147523, … |
| maximal normal subgroup of | #147325, #147326, #147523, … |
| maximal quotient of | #181944, #181949, #181956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13385);