Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10,16,11)(12,20,22,24)(13,14,18,15)(17,23,19,21);(1,3)(2,5)(4,7)(6,8)(9,11,14,18,16,10,15,13)(12,17,21,20,22,19,23,24);(3,7)(5,8)(9,12)(10,17)(11,19)(13,20)(14,21)(15,23)(16,22)(18,24);(9,13,14,11,16,18,15,10)(12,20,21,19,22,24,23,17)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27676, #6457242, #6457243, … |
| maximal normal subgroup of | C2×#27676, #6457242, #6457243, … |
| maximal quotient of | #314765, #314766, #314767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20664);