Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10611 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #396 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #126 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
138 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,9,13,8,4,15,14,16)(17,18,22,25)(19,26,24,21)(20,27,30,28)(23,31,32,29);(1,3,5,13,7,4,6,14)(2,8,10,9,11,16,12,15)(17,19,21,18,22,24,26,25)(20,28,29,23,30,27,31,32);(1,4,7,3)(2,9,11,15)(5,13,6,14)(8,12,16,10)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,31,26,29)Polycyclic presentation:
2^8|a^2=de,b^2=d,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fh,d^c=dg,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10613, #242363, #242364, … |
| maximal normal subgroup of | C2×#10613, #242363, #242364, … |
| maximal quotient of | #105187, #105191, #105193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3601);