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 | |
725 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)(3,6,8,4)(11,15)(14,16)(17,18,22,23)(19,24,27,31)(20,25,26,21)(28,29,30,32);(1,3,5,8)(2,6,7,4)(9,10,13,12)(11,14,15,16)(18,23)(19,27)(21,25)(28,30);(2,7)(3,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,24)(20,30)(21,29)(22,27)(23,31)(25,32)(26,28);(1,3)(2,6)(4,7)(5,8)(17,20,22,26)(18,25,23,21)(19,28,27,30)(24,29,31,32);(1,4,5,6)(2,3,7,8)(9,12,13,10)(11,16,15,14)(17,21,22,25)(18,26,23,20)(19,29,27,32)(24,28,31,30)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=f,e^2=fg,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×#49204, #7704772, #7704972, … |
| maximal normal subgroup of | C2×#49204, #7704772, #7704972, … |
| maximal quotient of | #560682, #561063, #561802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42192);