Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
703 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)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(25,26,30,29)(27,33,35,31)(28,36,38,32)(34,39,37,40);(1,2,4,6)(3,5,7,8)(10,11)(15,17)(16,18)(22,23)(26,29)(31,33)(32,36)(39,40);(1,3,4,7)(2,5,6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,30,35)(26,31,29,33)(28,37,38,34)(32,40,36,39);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,38)(26,32,29,36)(27,34,35,37)(31,39,33,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,26)(27,33,35,31)(28,36,38,32)(34,40,37,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132464, #253880, #253882, … |
| maximal normal subgroup of | #132464, #253880, #253882, … |
| maximal quotient of | #544545, #545120, #545416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34427);