Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C23×#12067 (order 2048) · Inn(G) = D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D16 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| socle, prime radical | C22 | #357 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(10,15)(11,19)(12,21)(13,25)(16,31)(17,34)(18,30)(20,24)(22,38)(23,28)(26,32)(27,29)(33,39)(35,36)(37,40);(9,10,14,18)(11,20,23,37)(12,24,26,40)(13,27,25,17)(15,28,30,19)(16,32,33,21)(22,39,35,31)(29,36,34,38)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=eg,d^2=f,e^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ce,e^b=eg,g^b=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7384, #150056, #150057, … |
| maximal normal subgroup of | C2×#7384, #150056, #150057, … |
| maximal quotient of | #93748, #93763, #93916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 372);