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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
711 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)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,31,34)(27,32,28,29)(30,38,39,33)(35,37,36,40);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(16,20)(19,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(2,5)(6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,29,31,32)(26,28,34,27)(30,37,39,40)(33,36,38,35);(1,3)(2,6)(4,7)(5,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,d^2=fg,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×#52619, #8092280, #8092472, … |
| maximal normal subgroup of | C2×#52619, #8092280, #8092472, … |
| maximal quotient of | #582297, #582520, #582566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45607);