Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(17,18,22,23)(19,24,27,30)(20,25,29,32)(21,28,31,26);(3,7)(6,8)(9,10,13,14)(11,12,15,16)(18,23)(19,27)(20,29)(21,31);(2,5)(6,8)(10,14)(11,15)(17,19)(18,24)(20,26)(21,25)(22,27)(23,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,25)(19,28)(21,30)(22,29)(23,32)(24,31)(26,27);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,21,22,31)(18,26,23,28)(19,25,27,32)(20,30,29,24)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132776, #487527, #488260, … |
| maximal normal subgroup of | #132776, #487527, #488260, … |
| maximal quotient of | #585373, #586082, #587644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47842);