Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | Aut(G) = C23×#45355 (order 2048) · Inn(G) = #386 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3214 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
221 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 221 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18,22,24,21,20,26,29)(19,25,27,32,28,23,30,31);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19,26,27,21,28,22,30)(18,23,24,32,20,25,29,31);(2,8)(4,5)(7,15)(11,13)(17,20)(18,21)(19,25)(22,24)(23,28)(26,29)(27,31)(30,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9371, #200168, #200169, … |
| maximal normal subgroup of | C2×#9371, #200168, #200169, … |
| maximal quotient of | #100043, #100049, #100051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2359);