Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
627 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 627 conjugacy classes of subgroups — too many to draw. Only its 175 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 — 3 generators on 20 points:
(5,6)(7,14,16,10)(8,13)(9,12)(11,15,19,17)(18,20);(1,2,3,4)(5,7)(6,10)(8,15)(9,16)(11,13)(12,14)(17,18)(19,20);(5,7)(6,11)(8,17)(9,16)(10,13)(12,19)(14,20)(15,18)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^b=fh,e^c=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129781, #129782, #129783, … |
| maximal normal subgroup of | #129781, #129782, #129783, … |
| maximal quotient of | #105640, #105641, #105642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3763);