Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C2×C4×#6342371 (order 4096) · Inn(G) = C23 · Out(G) = C22×C4×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C23 |
| commutator, socle | C23 | C22×C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
219 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,17,14,11,18,20,12,16,21,24,22,19,15,23);(2,4)(9,11,13,20,14,16,18,24,12,19,21,23,22,10,15,17);(6,8)(10,16)(11,19)(17,24)(20,23)Polycyclic presentation:
2^8|a^2=f,b^2=f,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #93731, #135950, #135951, … |
| maximal normal subgroup of | #93731, #135950, #135951, … |
| maximal quotient of | #135811, #135824, #135835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5901);