Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(12,20)(16,21)(18,24)(22,23)(25,26,31,30)(27,33,28,32)(29,34,38,37)(35,40,36,39);(2,5)(6,8)(10,14)(15,17)(16,21)(22,23)(25,27,31,28)(26,32,30,33)(29,36,38,35)(34,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,28,31,27)(26,33,30,32)(29,36,38,35)(34,40,37,39);(1,3)(2,6)(4,7)(5,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,29,31,38)(26,34,30,37)(27,35,28,36)(32,39,33,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=gh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133027, #487145, #487273, … |
| maximal normal subgroup of | #133027, #487145, #487273, … |
| maximal quotient of | #582186, #582411, #582635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45905);