Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 667 conjugacy classes of subgroups — too many to draw. Only its 179 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 — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,10,13,17)(8,15)(9,11)(12,16)(14,20,19,18);(5,7,8,14,9,13,16,19)(6,10,12,18,11,17,15,20);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105760, #106080, #106182, … |
| maximal normal subgroup of | #105760, #106080, #106182, … |
| maximal quotient of | #105711, #105712, C22⋊#1308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3725);