Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = #99 · Out(G) = C26 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #99 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | C2×#95 | C22 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C2×#95 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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 — 2 generators on 24 points:
(2,5,6,7)(9,10,11,13,12,14,15,16)(17,18)(19,22,24,21)(20,23);(1,2)(3,7)(4,6)(5,8)(18,21,23,22)Polycyclic presentation:
2^8|a^2=d,b^2=eh,c^2=fh,d^2=g,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7123, #124089, #124090, … |
| maximal normal subgroup of | C2×#7123, #124089, #124090, … |
| maximal quotient of | #92872, #92876, #92880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 111);