Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 3145728 |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C2 | C2×C43 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,7)(13,14,17,15)(16,19,20,18);(9,10,12,11)(14,15)(18,19);(1,2,3,4)(5,7,8,6)(9,11,12,10)(13,15,17,14)(16,19,20,18);(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=e,b^a=be,d^a=de,c^b=ce136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6738);