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 | |
737 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,20,27,24)(21,29,30,25)(26,31,32,28);(1,2,4,6)(3,5,8,7)(11,16)(13,15)(18,23)(20,24)(21,30)(26,32);(9,11)(10,13)(12,16)(14,15)(17,19,22,27)(18,24,23,20)(21,26,30,32)(25,31,29,28);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32);(1,3,4,8)(2,5,6,7)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=gh,d^2=gh,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 | #132738, #487454, #488113, … |
| maximal normal subgroup of | #132738, #487454, #488113, … |
| maximal quotient of | #585523, #586613, #587487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51741);