Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | Aut(G) = C23×#21452 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| 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 |
| prime radical | #2184 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 181 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)(10,17)(12,27)(13,30)(14,24)(15,25)(16,31)(18,36)(19,37)(20,22)(21,35)(23,29)(26,40)(28,39)(32,34)(33,38)Polycyclic presentation:
2^8|c^2=gh,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×#12166, #484636, #484639, … |
| maximal normal subgroup of | C2×#12166, #484636, #484639, … |
| maximal quotient of | #123173, #123176, #123179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5154);