Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | Aut(G) order 32768 |
| 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 | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
727 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 727 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,15)(8,11)(9,13)(10,14)(12,18)(16,20)(17,19);(6,11)(7,16)(8,9)(12,15)(13,14)(19,20);(2,4)(5,7,8,17,10,16,9,18)(6,12,13,20,14,19,11,15);(7,16)(12,19)(15,20)(17,18)Polycyclic presentation:
2^8|c^2=e,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 | #394600, #394632, #394633, … |
| maximal normal subgroup of | #394600, #394632, #394633, … |
| maximal quotient of | #238990, #238991, #238994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16519);