Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(28,36)(31,38)(34,40)(37,39);(10,15)(11,17)(16,22)(18,23)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2,4,5)(3,6,7,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6486197, #6486198, #6486207, … |
| maximal normal subgroup of | #6486197, #6486198, #6486207, … |
| maximal quotient of | #556435, #556437, #556451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30568);