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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
761 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)(3,7)(4,6)(5,8)(11,15)(14,16)(17,18,21,22)(19,23,25,29)(20,24,28,30)(26,31,27,32);(1,2,5,8)(3,6,4,7)(9,10,13,12)(11,14,15,16)(18,22)(19,25)(24,30)(26,27);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,27)(21,25)(22,29)(24,32)(26,28)(30,31);(1,2)(3,6)(4,7)(5,8)(9,10,13,12)(11,14,15,16)(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,16,15,14)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fh,d^2=fg,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×#50542, #7705817, #7706234, … |
| maximal normal subgroup of | C2×#50542, #7705817, #7706234, … |
| maximal quotient of | #560813, #561098, #561902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43530);