Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19504 |
| 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 |
| prime radical | #3212 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18)(19,23)(20,25)(21,30)(22,26)(24,27)(28,32)(29,31);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,30)(25,31)(26,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,32)(26,31);(3,9)(7,14)(10,15)(12,16)(17,20,22,21)(18,25,26,30)(19,27,28,29)(23,24,32,31)Polycyclic presentation:
2^8|b^2=e,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,d^c=dg,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19536, #6354630, #6354640, … |
| maximal normal subgroup of | C2×#19536, #6354630, #6354640, … |
| maximal quotient of | #189613, #197926, #198011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12524);