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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,13,18,14,19,29,16)(11,20,23,31,24,34,21,15)(12,25,26,33,28,32,27,17)(22,30,36,39,38,40,37,35);(2,5)(6,7)(9,11,14,24)(10,15,19,31)(12,22,28,38)(13,21,29,23)(16,20,18,34)(17,30,33,40)(25,35,32,39)(26,37,27,36);(10,16)(11,21)(13,29)(17,25)(18,19)(20,34)(22,36)(23,24)(26,27)(32,33)(35,39)(37,38);(1,3,4,8)(2,6,5,7)(9,12,14,28)(10,17,19,33)(11,22,24,38)(13,27,29,26)(15,30,31,40)(16,25,18,32)(20,35,34,39)(21,36,23,37)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh,e^d=eh,f^d=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26922, #6439732, #6439735, … |
| maximal normal subgroup of | C2×#26922, #6439732, #6439735, … |
| maximal quotient of | #312032, #312033, #312034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19910);