Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
382 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)(3,7)(4,5)(6,8)(9,10,14,19,13,18,15,20)(11,21,24,36,23,16,30,32)(12,25,28,34,27,35,29,17)(22,38,26,31,39,33,40,37);(2,5)(6,7)(9,11)(10,16)(12,26)(13,23)(14,24)(15,30)(17,33)(18,21)(19,32)(20,36)(22,29)(25,37)(27,40)(28,39)(31,35)(34,38);(9,12,13,27)(10,17,18,34)(11,22,23,39)(14,29,15,28)(16,31,21,37)(19,35,20,25)(24,40,30,26)(32,38,36,33);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,34)(19,20)(25,35)(31,37)(32,36)(33,38)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24675, #6449004, #6449006, … |
| maximal normal subgroup of | C2×#24675, #6449004, #6449006, … |
| maximal quotient of | #275808, #275814, #275820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17663);