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 |
| prime radical | #121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,13,14)(4,8,7,16)(5,12,15,9)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(30,31);(1,3,5,8,6,13,15,16)(2,7,9,11,10,4,12,14)(18,23)(24,31)(25,26)(27,32);(1,4)(2,8)(3,12)(5,14)(6,7)(9,13)(10,16)(11,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(3,13)(4,7)(8,16)(11,14)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=eh,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×#19381, #6354072, #6354074, … |
| maximal normal subgroup of | C2×#19381, #6354072, #6354074, … |
| maximal quotient of | #189131, #197834, #197835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12369);