Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | Aut(G) = C2×#72311718 (order 4096) · Inn(G) = #447 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3150 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(2,3)(5,6)(7,11)(8,15)(9,19)(10,14)(12,16)(13,17)(18,20);(1,2)(3,4)(5,6)(7,15)(8,11)(9,19)(10,14)(12,18)(13,17)(16,20);(1,3,4,2)(6,11)(7,16)(8,18)(9,17)(12,13)(14,15)(19,20)Polycyclic presentation:
2^8|c^2=d,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,c^b=cf,e^b=egh,f^b=fg,g^b=gh,e^c=eg,f^c=fg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125674, #125675, #125676, … |
| maximal normal subgroup of | #125674, #125675, #125676, … |
| maximal quotient of | #144886, #144887, #144888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6398);