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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
371 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)(3,4)(5,7)(6,8)(9,10)(11,21)(12,26)(13,19)(14,17)(15,20)(16,24)(18,30)(22,33)(23,38)(25,32)(27,40)(28,34)(29,36)(31,37)(35,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,27)(13,24)(14,25)(15,22)(17,33)(18,35)(19,21)(20,32)(23,28)(26,38)(29,39)(30,37)(31,36)(34,40);(1,4,5,6)(2,3,7,8)(10,17)(11,22)(14,15)(18,34)(19,20)(23,37)(24,25)(26,36)(27,39)(29,30)(32,33)(38,40);(2,7)(3,8)(9,12)(10,18)(11,23)(13,28)(14,30)(15,29)(16,31)(17,34)(19,36)(20,26)(21,35)(22,37)(24,27)(25,39)(32,40)(33,38)Polycyclic presentation:
2^8|c^2=eh,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 | #6669046, #6669049, #6669063, … |
| maximal normal subgroup of | #6669046, #6669049, #6669063, … |
| maximal quotient of | #335611, #335614, #335615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23611);