Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | |
673 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,3,6,5)(2,7,4,8)(10,12)(15,17)(16,20)(22,23)(26,29)(27,33)(28,36)(39,40);(1,4,6,2)(3,7,5,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,33)(26,31,29,35)(28,37,36,34)(32,40,38,39);(1,5,6,3)(2,7,4,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28,30,36)(26,32,29,38)(27,34,33,37)(31,39,35,40);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,30,26)(27,35,33,31)(28,38,36,32)(34,40,37,39)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#54355, #8103866, #8103892, … |
| maximal normal subgroup of | C2×#54355, #8103866, #8103892, … |
| maximal quotient of | #586026, #586148, #586629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47343);