Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
179 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 69 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,19,22,26,23,27,25,18,15,16,20,17,21,28,24);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,18,14)(16,24,19,25)(17,21,26,23)(20,27,22,28);(2,4)(13,14,18,15)(16,25,19,24)(17,23,26,21)(20,28,22,27)Polycyclic presentation:
2^8|a^2=dh,b^2=degh,c^2=h,d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dg,e^c=eg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #126288, #126289, #126314, … |
| maximal normal subgroup of | #126288, #126289, #126314, … |
| maximal quotient of | #146195, #146212, #146215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6519);