Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 111 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)(11,15,20,17)(18,19);(1,2)(3,4)(5,7)(6,10)(8,17)(9,16)(11,19)(12,14)(13,20)(15,18);(2,4)(5,7)(6,11,12,20)(8,15)(9,16)(10,13,14,19)(17,18)Polycyclic presentation:
2^8|a^2=g,c^2=d,e^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=eh,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133494, #133495, #133496, … |
| maximal normal subgroup of | #133494, #133495, #133496, … |
| maximal quotient of | #118668, #118669, #118710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4142);