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) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #424 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 139 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) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(18,24)(19,28)(20,31)(21,22)(25,39)(26,43)(27,38)(29,45)(30,35)(32,36)(33,34)(37,40)(41,48)(42,46);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18,23,27)(19,25,30,42)(20,32,34,43)(21,26,22,36)(24,33,38,31)(28,40,35,41)(29,46,45,39)(37,47,48,44);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(17,19)(18,25)(20,29)(21,35)(22,28)(23,30)(24,37)(26,41)(27,42)(31,44)(32,46)(33,47)(34,45)(36,40)(38,48)(39,43)Polycyclic presentation:
2^8|a^2=g,b^2=h,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×#12296, #484968, #484970, … |
| maximal normal subgroup of | C2×#12296, #484968, #484970, … |
| maximal quotient of | #126322, #126326, #126344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5284);