Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) = C23×#495725 (order 4096) · Inn(G) = C2×D8 · Out(G) = C25×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle, prime radical | C22 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,13,17)(11,19,21,38)(12,23,26,37)(14,28,25,18)(15,30,31,20)(16,33,35,27)(22,39,29,34)(24,36,40,32);(2,6)(3,7)(9,11,13,21)(10,15,17,31)(12,24,26,40)(14,29,25,22)(16,34,35,39)(18,36,28,32)(19,37,38,23)(20,33,30,27);(1,3,5,6,4,7,8,2)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=eh,d^2=fh,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=cg,d^b=df,f^b=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #93763, #93765, #151108, … |
| maximal normal subgroup of | #93763, #93765, #151108, … |
| maximal quotient of | #128730, #129153, #135920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5071);