Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,20)(16,19)(17,18);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,17)(14,18,19,20);(2,4)(5,8,9,7)(6,10,11,12)(14,18)(15,17)(19,20);(6,11)(10,12)(14,19)(18,20)Polycyclic presentation:
2^8|b^2=e,c^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101414, #101430, #101619, … |
| maximal normal subgroup of | #101414, #101430, #101619, … |
| maximal quotient of | #256292, #256295, #256298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17049);