Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13371 |
| automorphism | Aut(G) = C24×#3109 (order 2048) · Inn(G) = #476 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #476 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(1,2,3,4)(6,7,15,16)(8,17)(9,10)(11,18)(13,14,19,20);(5,6,8,7,11,14,17,13,12,15,10,16,18,20,9,19);(2,4)(5,7)(6,10)(8,15)(9,20)(11,19)(12,16)(13,18)(14,17)Polycyclic presentation:
2^8|a^2=de,b^2=d,d^2=g,e^2=gh,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fg,g^a=gh,c^b=cf,e^b=eh,f^b=fh,d^c=dg,e^c=eg,f^c=fgh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13467, #499304, #499305, … |
| maximal normal subgroup of | C2×#13467, #499304, #499305, … |
| maximal quotient of | #145909, #145919, #145950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6455);