Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,12)(11,15,17,19)(13,16,21,20)(18,22,23,24)(25,26,30,31)(27,34,32,28)(29,33,37,38)(35,40,39,36);(1,3,5,2)(4,7,8,6)(11,17)(13,21)(15,19)(16,20)(26,31)(27,32)(33,38)(35,39);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40);(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,33,31,38)(27,35,32,39)(28,36,34,40)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=g,d^2=fh,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 | C2×#42209, #7640690, #7640701, … |
| maximal normal subgroup of | C2×#42209, #7640690, #7640701, … |
| maximal quotient of | #544581, #545090, #545465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35197);