Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19733 |
| 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 |
| prime radical | #3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
352 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3,4,13,6,12,5,11)(2,7,14,15,9,16,8,10)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,2)(3,7)(4,14)(5,8)(6,9)(10,11)(12,16)(13,15)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(18,23)(22,29)(24,30)(28,32)Polycyclic presentation:
2^8|b^2=eg,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,d^c=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19740, #6357385, #6357388, … |
| maximal normal subgroup of | C2×#19740, #6357385, #6357388, … |
| maximal quotient of | #191528, #206277, #206280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12728);