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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
669 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)(9,10,14,15)(11,18,21,16)(12,17,22,13)(19,20,24,23)(25,26,29,33)(27,31,35,30)(28,32,36,39)(34,38,40,37);(1,2)(3,6)(4,5)(7,8)(10,15)(12,22)(16,18)(19,24)(25,27,29,35)(26,30,33,31)(28,34,36,40)(32,37,39,38);(2,5)(6,8)(9,11,14,21)(10,16,15,18)(12,19,22,24)(13,20,17,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,19)(13,15)(14,22)(16,23)(18,20)(21,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,17)(10,12,15,22)(11,20,21,23)(16,19,18,24)(25,28,29,36)(26,32,33,39)(27,34,35,40)(30,37,31,38)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=g,e^2=f,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×#52436, #8092438, #8092465, … |
| maximal normal subgroup of | C2×#52436, #8092438, #8092465, … |
| maximal quotient of | #582282, #582467, #582477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45424);