Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#553 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6,10,8)(7,12,9,11)(13,14,17,20)(15,18,16,19);(6,8)(11,12)(13,15,17,16)(14,18,20,19);(5,7,10,9)(6,11,8,12)(13,14)(15,18)(16,19)(17,20);(5,8,10,6)(7,12,9,11);(1,3,4,2)(5,9,10,7)(6,12,8,11)(13,16,17,15)(14,19,20,18)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #6384195, #6384196, #6384211, … |
| maximal normal subgroup of | #6384195, #6384196, #6384211, … |
| maximal quotient of | #513624, #513637, #513654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27289);