Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6,7,8)(9,10,13,18,15,19,14,29)(11,20,23,38,21,37,30,34)(12,24,26,35,28,36,27,17)(16,25,32,39,31,40,33,22);(1,2)(3,6)(4,8)(5,7)(9,11,15,21)(10,16,19,31)(12,25,28,40)(13,23,14,30)(17,34,35,38)(18,32,29,33)(20,36,37,24)(22,26,39,27);(1,3,5,4)(2,6,7,8)(11,21)(16,31)(20,37)(22,39)(23,30)(25,40)(32,33)(34,38)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=fgh,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,d^a=df,f^a=fh,g^a=gh,c^b=ch,d^b=dg,e^b=efgh,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129016, #129034, #131665, … |
| maximal normal subgroup of | #129016, #129034, #131665, … |
| maximal quotient of | #124112, #124221, #124234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4740);