Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,8)(9,12)(10,11)(14,19)(15,20);(2,4)(6,11)(7,9)(13,14)(15,16)(17,20)(18,19);(1,2)(3,4)(5,7,10,9)(6,8,11,12)(13,15,18,20)(14,16,19,17);(2,4)(5,8)(6,7)(9,11)(10,12)(13,16,18,17)(14,15,19,20);(1,2)(3,4)(5,9,10,7)(6,12,11,8)(13,17,18,16)(14,20,19,15)Polycyclic presentation:
2^8|c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#36138, #7629773, #7629777, … |
| maximal normal subgroup of | C2×#36138, #7629773, #7629777, … |
| maximal quotient of | #519172, #519176, #519188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29126);