Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24)(25,26)(27,31)(28,35)(29,32)(30,33)(34,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(10,11)(12,19)(16,18)(20,24)(25,27,30,35)(26,31,33,28)(29,34,37,40)(32,38,39,36);(2,5)(6,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,37)(26,32,33,39)(27,34,35,40)(28,36,31,38)Polycyclic presentation:
2^8|b^2=fg,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52651, #8092253, #8092473, … |
| maximal normal subgroup of | C2×#52651, #8092253, #8092473, … |
| maximal quotient of | #582277, #582464, #582486, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45639);