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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(14,16)(17,18,22,27)(19,24,30,23)(20,25,26,21)(28,29,31,32);(1,2)(3,7)(4,5)(6,8)(17,19,22,30)(18,23,27,24)(20,31,26,28)(21,32,25,29);(2,5)(6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,31)(21,29)(22,30)(23,27)(25,32)(26,28);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,20,22,26)(18,25,27,21)(19,28,30,31)(23,32,24,29);(1,3)(2,6)(4,8)(5,7)(17,21,22,25)(18,26,27,20)(19,29,30,32)(23,31,24,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fg,e^2=f,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×#46562, #7700468, #7700572, … |
| maximal normal subgroup of | C2×#46562, #7700468, #7700572, … |
| maximal quotient of | #576789, #576902, #577153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39550);