Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C27×#442 (order 8192) · Inn(G) = C2×D8 · Out(G) = C26×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 79 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:
(2,4)(5,6)(7,8)(9,10,11,15,19,23,24,22,13,17,20,21,12,16,18,14);(1,2,3,4)(10,14)(11,18)(12,19)(15,16)(17,22)(20,24)(21,23);(6,8)(10,14)(11,18)(12,19)(15,16)(17,22)(20,24)(21,23)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^b=dg,e^b=eg,f^b=fg,g^b=gh,d^c=dg,e^c=eg,f^c=fg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13012, #494497, #494498, … |
| maximal normal subgroup of | C2×#13012, #494497, #494498, … |
| maximal quotient of | #136685, #136719, #136893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6000);