Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,13)(14,16)(17,18,21,25)(19,23,27,22)(20,28,30,24)(26,31,29,32);(2,5)(6,8)(11,15)(14,16)(17,19,21,27)(18,22,25,23)(20,29,30,26)(24,32,28,31);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,32)(28,31)(29,30);(1,3)(2,6)(4,7)(5,8)(17,20,21,30)(18,24,25,28)(19,26,27,29)(22,31,23,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,32)(28,30)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | #98116, #179351, #179356, … |
| maximal normal subgroup of | #98116, #179351, #179356, … |
| maximal quotient of | #606743, #606792, #606797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46171);