Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19671 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8)(3,4,7,5)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(1,3,5,2,6,7,4,8)(11,19)(15,23)(18,24)(20,21);(9,11,13,21,14,19,12,20)(10,15,16,18,17,23,22,24);(1,4,6,5)(2,7,8,3)(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^b=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19694, #6357271, #6357272, … |
| maximal normal subgroup of | C2×#19694, #6357271, #6357272, … |
| maximal quotient of | #191450, #206192, #206194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12682);