Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(11,15)(14,16)(17,18)(19,20)(21,29)(22,23)(24,27)(25,30)(26,31)(28,32);(1,3)(2,6)(4,5)(7,8)(9,10,13,12)(11,14,15,16)(18,23)(20,24)(25,29)(28,31);(3,4)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,29)(30,32);(1,4,5,3)(2,6,7,8)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32);(1,2)(3,8)(4,6)(5,7)(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47353, #7701307, #7701448, … |
| maximal normal subgroup of | C2×#47353, #7701307, #7701448, … |
| maximal quotient of | #576968, #576989, #577094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40341);