Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
975 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 975 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20);(1,2)(3,4)(8,9)(12,17)(14,19)(15,18);(2,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(5,8)(6,12)(7,15)(9,10)(11,19)(13,17)(14,20)(16,18)Polycyclic presentation:
2^8|f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139733, #152715, #152717, … |
| maximal normal subgroup of | #139733, #152715, #152717, … |
| maximal quotient of | #245168, #245176, #245180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16900);