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×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 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)(3,6)(4,8)(5,7)(9,10)(11,21)(12,27)(13,30)(14,20)(15,19)(16,24)(17,29)(18,28)(22,35)(23,33)(25,37)(26,34)(31,36)(32,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11,14,25)(10,16,20,34)(12,23,29,40)(13,26,15,24)(17,35,27,38)(18,33,36,32)(19,37,30,21)(22,31,39,28);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,31)(16,32)(18,19)(20,27)(21,38)(23,24)(25,39)(26,40)(30,36)(33,34)(35,37);(9,12)(10,18)(11,23)(13,31)(14,29)(15,28)(16,33)(17,19)(20,36)(21,38)(22,24)(25,40)(26,39)(27,30)(32,34)(35,37)Polycyclic presentation:
2^8|b^2=f,c^2=fh,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×#26192, #6438603, #6438604, … |
| maximal normal subgroup of | C2×#26192, #6438603, #6438604, … |
| maximal quotient of | #306317, #306320, #306321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19180);