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 | #3403 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #446 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 160 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,15,19)(11,20,23,38)(12,17,14,29)(13,25,28,36)(16,31,33,22)(18,24,37,27)(21,39,30,35)(26,32,40,34);(3,7)(5,8)(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,32)(18,35)(19,33)(20,25)(22,27)(24,31)(28,40)(29,34)(36,38)(37,39);(1,2,4,6)(3,8,7,5)(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,27,28,24)(16,32,33,34)(18,36,37,25)(20,35,38,39)(22,40,31,26)Polycyclic presentation:
2^8|a^2=h,c^2=g,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×#12338, #489192, #489193, … |
| maximal normal subgroup of | C2×#12338, #489192, #489193, … |
| maximal quotient of | #125663, #125664, #125697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5326);