Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26,30,33)(27,34,35,31)(28,29,32,39)(36,37,38,40);(1,2,4,6)(3,5,7,8)(10,14)(12,21)(15,17)(18,24)(27,35)(28,32)(29,39)(31,34);(1,3,4,7)(2,5,6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,35)(26,31,33,34)(28,38,32,36)(29,37,39,40);(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,30,32)(26,29,33,39)(27,36,35,38)(31,40,34,37);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29)(26,32)(27,37)(28,33)(30,39)(31,36)(34,38)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=g,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132492, #253553, #253586, … |
| maximal normal subgroup of | #132492, #253553, #253586, … |
| maximal quotient of | #560390, #560532, #561573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44215);