Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9784 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D8 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
237 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 237 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,4,6,8,3)(9,10,13,17,14,18,12,15)(11,19,22,16,20,23,21,24);(1,3,5,6,4,7,8,2)(10,15,18,17)(11,20)(16,23,24,19)(21,22);(2,6)(3,7)(9,11,12,21,14,20,13,22)(10,16,17,23,18,24,15,19)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eg,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cg,d^b=dh,f^b=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #138862, #138863, #138874, … |
| maximal normal subgroup of | #138862, #138863, #138874, … |
| maximal quotient of | #102071, #102073, #102077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2779);