Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(7,8)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26,31,33)(27,34,28,30)(29,37,39,32)(35,38,36,40);(11,17)(13,21)(15,19)(16,20)(25,27,31,28)(26,30,33,34)(29,35,39,36)(32,38,37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28,31,27)(26,30,33,34)(29,35,39,36)(32,40,37,38);(1,3,5,4)(2,7,6,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,29,31,39)(26,32,33,37)(27,35,28,36)(30,38,34,40);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,27)(28,33)(29,38)(31,34)(32,35)(36,37)(39,40)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=f,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#42257, #7640638, #7640667, … |
| maximal normal subgroup of | C2×#42257, #7640638, #7640667, … |
| maximal quotient of | #544897, #545304, #545895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35245);