Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3219 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
408 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 109 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 — 4 generators on 48 points:
(2,7)(3,10)(4,5)(9,12)(11,13)(14,16)(17,18)(19,27)(20,24)(21,33)(22,25)(23,30)(26,36)(28,44)(29,41)(31,42)(32,39)(34,38)(35,40)(37,46)(43,45)(47,48);(1,2)(3,8)(4,12)(5,7)(6,9)(10,16)(11,15)(13,14)(17,19,22,29)(18,23,25,38)(20,28,32,45)(21,34,36,30)(24,37,39,48)(26,41,33,27)(31,47,35,46)(40,43,42,44);(1,3,6,11)(2,8,9,15)(4,10,5,13)(7,14,12,16)(17,20,22,32)(18,24,25,39)(19,28,29,45)(21,35,36,31)(23,37,38,48)(26,42,33,40)(27,43,41,44)(30,47,34,46);(2,9)(7,12)(8,15)(14,16)(18,25)(19,29)(24,39)(26,33)(28,45)(30,34)(40,42)(46,47)Polycyclic presentation:
2^8|b^2=h,c^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh31 conjugacy classes — view the full character table.
| maximal subgroup of | #250487, #250494, #250495, … |
| maximal normal subgroup of | #250487, #250494, #250495, … |
| maximal quotient of | #168923, #168927, #168929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12080);