Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C24 · Out(G) = C211 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
775 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,22)(19,21,23,24);(1,3,6,2)(4,7,5,8)(17,19)(18,21)(20,23)(22,24);(10,13)(14,16)(19,23)(21,24);(1,4,6,5)(2,7,3,8)(10,13)(14,16);(1,5,6,4)(2,7,3,8)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=h,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 | #6378859, #6378860, #6378861, … |
| maximal normal subgroup of | #6378859, #6378860, #6378861, … |
| maximal quotient of | #520736, #520738, #520858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45239);