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, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,30,33)(27,29,32,38)(28,31,35,39)(34,37,36,40);(1,2,4,6)(3,5,7,8)(10,12)(15,17)(16,20)(22,23)(27,32)(28,35)(29,38)(31,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,32)(26,29,33,38)(28,36,35,34)(31,40,39,37);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28)(26,31)(27,34)(29,37)(30,35)(32,36)(33,39)(38,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,33)(28,37)(30,38)(31,36)(34,39)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=fh,d^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 | #132145, #253452, #253498, … |
| maximal normal subgroup of | #132145, #253452, #253498, … |
| maximal quotient of | #560423, #560619, #561431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43177);