Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 65536 |
| 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 | #2828 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
778 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)(9,10,12,14)(11,13,15,16)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,32)(28,29);(1,2)(3,7)(4,5)(6,8)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,32,30,29);(2,5)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,19)(18,24,26,21)(22,28,31,27)(25,32,30,29);(1,2)(3,6)(4,5)(7,8)(17,21,23,24)(18,19,26,20)(22,29,31,32)(25,27,30,28);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,13,14,16)(17,22,23,31)(18,25,26,30)(19,27,20,28)(21,29,24,32)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=f,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 | #130420, #486066, #486560, … |
| maximal normal subgroup of | #130420, #486066, #486560, … |
| maximal quotient of | #576874, #577140, #577367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39963);