Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | Aut(G) = C23×#21452 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | C2×D4 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #436 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 39 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:
(1,2,6,10)(3,11,14,15)(4,9,5,8)(7,13,16,12)(18,23)(19,22)(20,27)(24,31)(25,32)(26,29)(28,30);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,18,21,25)(19,24,22,30)(20,28,29,31)(23,26,32,27);(2,8)(4,5)(7,15)(9,10)(11,16)(12,13)(17,19,21,22)(18,24,25,30)(20,27,29,26)(23,31,32,28)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=h,d^2=fh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12403, #489102, #489104, … |
| maximal normal subgroup of | C2×#12403, #489102, #489104, … |
| maximal quotient of | #125896, #125898, #125912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5391);