Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
187 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(5,8)(9,10)(11,22)(12,16)(13,20)(14,19)(15,21)(17,24)(18,26)(23,36)(25,37)(27,34)(28,32)(29,33)(30,35)(31,40)(38,39);(1,2)(3,5)(4,6)(7,8)(10,16)(12,26)(13,14)(17,31)(18,20)(19,32)(21,33)(22,36)(23,39)(24,30)(27,40)(28,29)(34,37)(35,38);(1,3,4,7)(2,5,6,8)(9,11,15,25)(10,17,21,35)(12,27,29,23)(13,30,14,24)(16,31,33,38)(18,36,32,34)(19,37,20,22)(26,40,28,39)Polycyclic presentation:
2^8|c^2=g,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=eh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12164, #484636, #484637, … |
| maximal normal subgroup of | C2×#12164, #484636, #484637, … |
| maximal quotient of | #123171, #123174, #123177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5152);