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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
844 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)(3,6)(4,5)(7,8)(17,18)(19,27)(20,25)(21,31)(22,23)(24,28)(26,32)(29,30);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,12,15,16)(18,23)(19,28)(20,30)(21,32);(2,5)(6,7)(10,14)(11,15)(17,19)(18,24)(20,31)(21,29)(22,28)(23,27)(25,32)(26,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,30)(18,25,23,29)(19,26,28,31)(21,27,32,24);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,16)(14,15)(17,21)(18,26)(19,29)(20,27)(22,32)(23,31)(24,30)(25,28)Polycyclic presentation:
2^8|b^2=gh,d^2=f,e^2=fg,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×#47175, #7701124, #7701147, … |
| maximal normal subgroup of | C2×#47175, #7701124, #7701147, … |
| maximal quotient of | #577001, #577017, #577134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40163);