Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | #2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
780 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)(3,5)(4,7)(6,8)(11,15)(14,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31);(2,3)(6,7)(10,13)(14,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)Polycyclic presentation:
2^8|b^2=fg,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53290, #8105685, #8105769, … |
| maximal normal subgroup of | C2×#53290, #8105685, #8105769, … |
| maximal quotient of | #606740, #606774, #606790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46278);