Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
209 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,15,18)(16,19,22,26)(17,25,23,21)(20,28,27,24);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,18)(19,26)(20,27)(21,25);(2,4)(14,19)(16,24)(17,23)(18,26)(20,21)(22,28)(25,27)Polycyclic presentation:
2^8|a^2=d,b^2=dh,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^c=eg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13365, #498949, #498951, … |
| maximal normal subgroup of | C2×#13365, #498949, #498951, … |
| maximal quotient of | #144446, #144478, #144517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6353);