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) = C29 |
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 | |
1651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(9,10)(11,12)(13,15)(14,16);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16);(2,4)(6,8);(2,4)(10,15)(11,16);(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^8|e^2=f,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 | #131260, #484914, #485479, … |
| maximal normal subgroup of | #131260, #484914, #485479, … |
| maximal quotient of | #520663, #520713, #520724, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45084);