Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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×#562 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1112 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)(6,8)(10,11)(14,17)(15,19);(5,6,9,8)(7,11,12,10)(14,17)(18,20);(1,2,4,3)(13,14)(15,18)(16,17)(19,20);(5,7,9,12)(6,10,8,11)(13,15)(14,18)(16,19)(17,20);(1,3,4,2)(5,8,9,6)(7,11,12,10)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#34519, #7626149, #7626155, … |
| maximal normal subgroup of | C2×#34519, #7626149, #7626155, … |
| maximal quotient of | #513870, #513872, #513873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27507);