Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
431 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)(9,10,13,19)(11,21,23,16)(12,25,27,37)(14,30,15,20)(17,29,36,28)(18,24,33,31)(22,34,39,35)(26,38,40,32);(1,3,4,8)(2,5,6,7)(9,11,13,23)(10,16,19,21)(12,26,27,40)(14,24,15,31)(17,34,36,35)(18,20,33,30)(22,28,39,29)(25,32,37,38);(9,12,13,27)(10,17,19,36)(11,22,23,39)(14,29,15,28)(16,32,21,38)(18,35,33,34)(20,37,30,25)(24,40,31,26);(2,6)(3,8)(9,11)(10,18)(12,22)(13,23)(14,31)(15,24)(16,30)(17,35)(19,33)(20,21)(25,32)(26,29)(27,39)(28,40)(34,36)(37,38)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #6668398, #6668399, #6668406, … |
| maximal normal subgroup of | #6668398, #6668399, #6668406, … |
| maximal quotient of | #335082, #335083, #335084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23541);