Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,15)(14,16)(17,18,20,22)(19,23,24,21)(27,31)(29,32);(2,5)(3,7)(10,11)(14,15)(19,24)(21,23)(25,26,28,30)(27,32,31,29);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,27,28,31)(26,29,30,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,21)(20,24)(22,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17303, #845192, #845200, … |
| maximal normal subgroup of | C2×#17303, #845192, #845200, … |
| maximal quotient of | #156569, #156587, #156588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10291);