Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,4,7)(10,15)(11,16)(17,18,22,23)(19,20,25,24)(21,26,30,31)(27,28,32,29);(1,3,5,4)(2,6,8,7)(9,10)(11,12)(13,16)(14,15)(18,23)(19,25)(26,31)(27,32);(1,2)(3,6)(4,7)(5,8)(9,11,14,16)(10,12,15,13)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,11,15,16)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32);(1,4,5,3)(2,7,8,6)(9,13,14,12)(10,16,15,11)(17,21,22,30)(18,26,23,31)(19,27,25,32)(20,28,24,29)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,d^2=g,e^2=g,b^a=bf,c^a=cg,c^b=ch,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6580776, #6580781, #6580841, … |
| maximal normal subgroup of | #6580776, #6580781, #6580841, … |
| maximal quotient of | #517955, #517957, #517962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29046);