Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,26,30,34)(27,35,36,31)(28,32,38,40)(29,33,39,37);(1,2,4,6)(3,5,7,8)(10,14)(11,17)(12,21)(22,24)(27,36)(29,39)(31,35)(33,37);(1,3,4,7)(2,5,6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,36)(26,31,34,35)(28,37,38,33)(29,32,39,40);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,28)(26,32)(27,33)(29,31)(30,38)(34,40)(35,39)(36,37);(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,29)(26,33)(27,32)(28,31)(30,39)(34,37)(35,38)(36,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133808, #251960, #251987, … |
| maximal normal subgroup of | #133808, #251960, #251987, … |
| maximal quotient of | #576669, #576670, #576770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40213);