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) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D16 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C22 | #357 |
| prime radical | C2×D16 | C4 |
| 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 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,15)(12,24)(13,27)(14,19)(16,20)(17,21)(18,25)(22,34)(23,30)(26,40)(28,36)(29,35)(31,39)(32,37)(33,38);(10,15)(11,20)(12,21)(13,25)(16,17)(18,34)(19,30)(22,39)(23,37)(24,28)(26,38)(27,29)(31,40)(32,33)(35,36)Polycyclic presentation:
2^8|a^2=d,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×#7383, #150056, #150057, … |
| maximal normal subgroup of | C2×#7383, #150056, #150057, … |
| maximal quotient of | #93747, #93762, #93915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 371);