Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14)(15,20)(16,19)(17,18);(6,7)(10,11)(13,15)(14,19)(16,18)(17,20);(1,2)(3,4)(15,16)(19,20);(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,19,17,20);(5,8,9,12)(6,10,7,11)(13,17,18,14)(15,19,16,20)Polycyclic presentation:
2^8|a^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46826, #7700665, #7700704, … |
| maximal normal subgroup of | C2×#46826, #7700665, #7700704, … |
| maximal quotient of | #576626, #576666, #576679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39814);