Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C23 |
| commutator | C23 | C2×C42 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 191 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) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,7,8)(10,11)(13,14,17,20,16,19,18,15);(9,10,12,11)(13,15,18,19,16,20,17,14);(2,4)(9,11,12,10)(14,19)(15,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10798, #353479, #353480, … |
| maximal normal subgroup of | C2×#10798, #353479, #353480, … |
| maximal quotient of | #108387, #108388, #108397, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3786);