Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,21)(12,16)(13,19)(14,20)(15,24)(17,26)(18,30)(22,35)(23,32)(25,33)(27,36)(28,37)(29,34)(31,38)(39,40);(9,11,14,25)(10,15,20,34)(12,23,28,38)(13,29,30,24)(16,35,37,40)(17,32,36,31)(18,33,19,21)(22,26,39,27);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,22)(13,26)(14,28)(15,31)(17,19)(18,36)(20,37)(21,35)(23,29)(24,38)(25,39)(27,30)(32,34)(33,40);(9,12)(10,17)(11,23)(13,27)(14,28)(15,32)(16,19)(18,37)(20,36)(21,35)(22,29)(24,39)(25,38)(26,30)(31,34)(33,40)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25985, #6438157, #6438159, … |
| maximal normal subgroup of | C2×#25985, #6438157, #6438159, … |
| maximal quotient of | #305248, #305249, #305250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18973);