Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = C25×#120 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #3171 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
226 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(2,5,6,8)(9,10)(11,18,19,22)(12,15,20,21)(13,16)(14,17)(23,24);(3,7)(5,8)(9,11,13,21,14,19,23,15)(10,12,16,22,17,20,24,18);(1,2)(3,8)(4,6)(5,7)(9,12)(10,15)(11,16)(13,22)(14,20)(17,21)(18,23)(19,24)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=g,e^2=gh,b^a=bd,c^a=ce,e^a=eg,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8997, #194424, #194425, … |
| maximal normal subgroup of | C2×#8997, #194424, #194425, … |
| maximal quotient of | #99535, #99536, #99537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1985);