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 | |
807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28);(1,2,5,3)(4,7,8,6)(9,10,13,14)(11,12,15,16)(18,23)(20,27)(25,29)(28,32);(2,3)(6,7)(10,14)(11,15)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,21,22,30)(18,25,23,29)(19,26,24,31)(20,28,27,32)Polycyclic presentation:
2^8|a^2=fg,b^2=h,c^2=f,d^2=g,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×#47077, #7701147, #7701165, … |
| maximal normal subgroup of | C2×#47077, #7701147, #7701165, … |
| maximal quotient of | #576942, #576981, #577056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40065);