Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,22)(20,24)(25,26)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(1,3)(2,5)(4,7)(6,8)(10,14)(12,19)(15,21)(17,20)(25,27)(26,30)(28,33)(29,35)(31,38)(32,37)(34,36)(39,40);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(3,7)(5,8)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(26,31)(30,38)(32,40)(37,39)Polycyclic presentation:
2^8|a^2=f,c^2=e,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#32844, #6631743, #6631750, … |
| maximal normal subgroup of | C2×#32844, #6631743, #6631750, … |
| maximal quotient of | #391310, #391312, #391336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25832);