Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(9,10,14,18)(11,15,20,30)(12,22,25,34)(13,26,28,17)(16,27,33,24)(19,35,23,40)(21,38,37,31)(29,39,32,36);(1,2,4,3)(5,6,7,8)(9,11)(10,15)(12,23)(13,21)(14,20)(16,32)(17,31)(18,30)(19,25)(22,40)(24,39)(26,38)(27,36)(28,37)(29,33)(34,35);(6,8)(9,12,14,25)(10,16,18,33)(11,19,20,23)(13,27,28,24)(15,29,30,32)(17,34,26,22)(21,39,37,36)(31,40,38,35);(1,3,4,2)(11,20)(15,30)(19,23)(21,37)(29,32)(31,38)(35,40)(36,39)Polycyclic presentation:
2^8|a^2=h,b^2=egh,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123172, #123182, #332624, … |
| maximal normal subgroup of | #123172, #123182, #332624, … |
| maximal quotient of | #317929, #317930, #317931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21249);