Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(5,7,10,6)(8,11,9,12);(2,4)(14,17)(18,20);(5,8,10,9)(6,11,7,12)(14,17)(18,20);(5,9,10,8)(6,11,7,12)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|b^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #97522, #165697, #165706, … |
| maximal normal subgroup of | #97522, #165697, #165706, … |
| maximal quotient of | #520736, #520737, #520782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45157);