Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 167 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)(3,4)(5,6)(7,15)(8,10)(9,13)(11,14)(12,17)(16,20)(18,19);(5,7)(6,12)(8,19)(9,16)(10,17)(11,18)(13,15)(14,20);(2,4)(5,8,9,14,11,13,10,6)(7,15,17,20,18,19,16,12);(5,9,11,10)(6,13,14,8)(7,16,18,17)(12,15,20,19)Polycyclic presentation:
2^8|c^2=e,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #394683, #394684, #395430, … |
| maximal normal subgroup of | #394683, #394684, #395430, … |
| maximal quotient of | #238993, #238997, #238998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16521);