Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#546 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(7,11)(10,12)(14,15)(18,19);(2,3)(5,6)(7,12)(8,9)(10,11)(13,14,17,15)(16,19,20,18);(6,9)(10,12)(13,15,17,14)(16,19,20,18);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,16)(14,18)(15,19)(17,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=g,d^2=fg,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7090212, #7090222, #7090228, … |
| maximal normal subgroup of | #7090212, #7090222, #7090228, … |
| maximal quotient of | #535407, #535508, #536064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38072);