Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(10,14)(15,18)(16,21)(22,24)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(10,14)(11,17)(12,20)(22,24)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,28,30,37)(26,32,33,40)(27,35,29,36)(31,38,34,39);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,29,30,27)(26,31,33,34)(28,35,37,36)(32,39,40,38);(1,3,4,7)(2,5,6,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,39)(36,38)(37,40)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=h,d^2=f,e^2=g,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 | C2×#50370, #7705857, #7706223, … |
| maximal normal subgroup of | C2×#50370, #7705857, #7706223, … |
| maximal quotient of | #560746, #561077, #561849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43358);