Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2)(3,6)(4,5)(7,8)(9,10,15,17,14,20,12,18)(11,16,24,32,25,33,22,30)(13,19,28,34,27,37,26,35)(21,36,38,29,39,40,23,31);(2,5)(6,8)(9,11)(10,16)(12,22)(13,21)(14,25)(15,24)(17,32)(18,30)(19,36)(20,33)(23,26)(27,39)(28,38)(29,34)(31,35)(37,40);(1,3,4,7)(2,6,5,8)(10,17)(11,21)(12,15)(13,26)(16,29)(18,20)(22,38)(23,24)(25,39)(27,28)(30,40)(31,33)(32,36)(34,35)Polycyclic presentation:
2^8|a^2=eh,c^2=d,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fg,g^b=gh,e^c=eh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13485, #499355, #499356, … |
| maximal normal subgroup of | C2×#13485, #499355, #499356, … |
| maximal quotient of | #146006, #146007, #146019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6473);