Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19368 |
| 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 | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(1,3,5,4)(2,6,7,8)(10,16)(11,17)(14,21)(18,23)Polycyclic presentation:
2^8|b^2=e,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,d^c=dg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19376, #6354064, #6354073, … |
| maximal normal subgroup of | C2×#19376, #6354064, #6354073, … |
| maximal quotient of | #189130, #197836, #197837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12364);