Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
331 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)(3,8)(4,6)(5,7)(9,10,16,20,14,19,15,31)(11,21,26,36,24,17,25,34)(12,27,30,35,29,39,13,18)(22,37,23,32,28,40,38,33);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,17,19,21)(12,28,29,22)(13,23,30,38)(15,25,16,26)(18,33,35,32)(20,34,31,36)(27,37,39,40);(9,12,15,13,14,29,16,30)(10,18,20,27,19,35,31,39)(11,22,25,38,24,28,26,23)(17,32,34,40,21,33,36,37);(1,4,5,3)(2,6,7,8)(9,13,15,29,14,30,16,12)(10,18,31,39,19,35,20,27)(11,23,25,22,24,38,26,28)(17,33,36,40,21,32,34,37)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=eg,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124924, #136784, #140860, … |
| maximal normal subgroup of | #124924, #136784, #140860, … |
| maximal quotient of | #213408, #213411, #213418, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14725);