Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,8,7)(10,12)(13,14,17,18)(15,19,20,16);(5,6,8,7)(14,18)(16,19);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19);(1,3,4,2)(5,7,8,6)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=eg,d^2=eg,b^a=be,c^a=cf,d^b=de100 conjugacy classes — view the full character table.
| maximal subgroup of | #219854, #219856, #219865, … |
| maximal normal subgroup of | #219854, #219856, #219865, … |
| maximal quotient of | #153725, #153727, #153729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6804);