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 | C2×#121 | 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)(3,6,8,4)(11,15)(14,16)(17,18)(19,20)(21,29)(22,23)(24,30)(25,28)(26,32)(27,31);(1,3,5,8)(2,6,7,4)(9,10,13,12)(11,14,15,16)(18,23)(19,25)(21,30)(27,32);(2,7)(3,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,25)(18,20,23,28)(21,26,30,31)(24,27,29,32);(1,3)(2,6)(4,7)(5,8)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,31)(30,32);(1,4,5,6)(2,3,7,8)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^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×#49187, #7704237, #7704972, … |
| maximal normal subgroup of | C2×#49187, #7704237, #7704972, … |
| maximal quotient of | #560673, #561045, #561794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42175);